id sid tid token lemma pos cana-1326 1 1 communications communication NOUN cana-1326 1 2 on on ADP cana-1326 1 3 applied apply VERB cana-1326 1 4 nonlinear nonlinear ADJ cana-1326 1 5 analysis analysis NOUN cana-1326 1 6 issn issn NOUN cana-1326 1 7 : : PUNCT cana-1326 1 8 1074 1074 NUM cana-1326 1 9 - - PUNCT cana-1326 1 10 133x 133x NUM cana-1326 1 11 vol vol NOUN cana-1326 1 12 31 31 NUM cana-1326 1 13 no no NOUN cana-1326 1 14 . . PUNCT cana-1326 2 1 7s 7 NOUN cana-1326 2 2 ( ( PUNCT cana-1326 2 3 2024 2024 NUM cana-1326 2 4 ) ) PUNCT cana-1326 2 5 466 466 NUM cana-1326 2 6 https://internationalpubls.com https://internationalpubls.com X cana-1326 2 7 common common ADJ cana-1326 2 8 fixed fix VERB cana-1326 2 9 point point NOUN cana-1326 2 10 results result NOUN cana-1326 2 11 for for ADP cana-1326 2 12 contractive contractive ADJ cana-1326 2 13 mappings mapping NOUN cana-1326 2 14 in in ADP cana-1326 2 15 bicomplex bicomplex NOUN cana-1326 2 16 valued value VERB cana-1326 2 17 b b X cana-1326 2 18 - - PUNCT cana-1326 2 19 metric metric ADJ cana-1326 2 20 spaces space NOUN cana-1326 2 21 md md PROPN cana-1326 2 22 . . PROPN cana-1326 2 23 azizul azizul ADJ cana-1326 2 24 hoque hoque PROPN cana-1326 2 25 department department PROPN cana-1326 2 26 of of ADP cana-1326 2 27 mathematics mathematics PROPN cana-1326 2 28 , , PUNCT cana-1326 2 29 sreegopal sreegopal ADJ cana-1326 2 30 banerjee banerjee PROPN cana-1326 2 31 college college PROPN cana-1326 2 32 , , PUNCT cana-1326 2 33 mogra mogra ADJ cana-1326 2 34 , , PUNCT cana-1326 2 35 dist dist NOUN cana-1326 2 36 - - PUNCT cana-1326 2 37 hooghly hooghly ADV cana-1326 2 38 , , PUNCT cana-1326 2 39 pin712148 pin712148 NOUN cana-1326 2 40 , , PUNCT cana-1326 2 41 west west PROPN cana-1326 2 42 bengal bengal PROPN cana-1326 2 43 , , PUNCT cana-1326 3 1 india india PROPN cana-1326 3 2 . . PUNCT cana-1326 3 3 email email NOUN cana-1326 3 4 : : PUNCT cana-1326 3 5 mhoque3@gmail.com mhoque3@gmail.com X cana-1326 3 6 article article NOUN cana-1326 3 7 history history NOUN cana-1326 3 8 : : PUNCT cana-1326 3 9 received receive VERB cana-1326 3 10 : : PUNCT cana-1326 3 11 01 01 NUM cana-1326 3 12 - - PUNCT cana-1326 3 13 06 06 NUM cana-1326 3 14 - - PUNCT cana-1326 3 15 2024 2024 NUM cana-1326 3 16 revised revise VERB cana-1326 3 17 : : PUNCT cana-1326 3 18 03 03 NUM cana-1326 3 19 - - PUNCT cana-1326 3 20 07 07 NUM cana-1326 3 21 - - PUNCT cana-1326 3 22 2024 2024 NUM cana-1326 3 23 accepted accept VERB cana-1326 3 24 : : PUNCT cana-1326 3 25 29 29 NUM cana-1326 3 26 - - SYM cana-1326 3 27 07 07 NUM cana-1326 3 28 - - PUNCT cana-1326 3 29 2024 2024 NUM cana-1326 3 30 abstract abstract NOUN cana-1326 3 31 : : PUNCT cana-1326 3 32 in in ADP cana-1326 3 33 this this DET cana-1326 3 34 article article NOUN cana-1326 3 35 , , PUNCT cana-1326 3 36 we we PRON cana-1326 3 37 extend extend VERB cana-1326 3 38 and and CCONJ cana-1326 3 39 generalised generalise VERB cana-1326 3 40 the the DET cana-1326 3 41 results result NOUN cana-1326 3 42 of of ADP cana-1326 3 43 ahmad ahmad PROPN cana-1326 3 44 et.al et.al PROPN cana-1326 3 45 . . PROPN cana-1326 3 46 , , PUNCT cana-1326 3 47 and and CCONJ cana-1326 3 48 to to PART cana-1326 3 49 establish establish VERB cana-1326 3 50 the the DET cana-1326 3 51 existence existence NOUN cana-1326 3 52 and and CCONJ cana-1326 3 53 uniqueness uniqueness NOUN cana-1326 3 54 of of ADP cana-1326 3 55 common common ADJ cana-1326 3 56 fixed fix VERB cana-1326 3 57 points point NOUN cana-1326 3 58 for for ADP cana-1326 3 59 pair pair NOUN cana-1326 3 60 of of ADP cana-1326 3 61 self self NOUN cana-1326 3 62 mappings mapping NOUN cana-1326 3 63 on on ADP cana-1326 3 64 a a DET cana-1326 3 65 closed closed ADJ cana-1326 3 66 ball ball NOUN cana-1326 3 67 in in ADP cana-1326 3 68 bicomplex bicomplex NOUN cana-1326 3 69 valued value VERB cana-1326 3 70 b b NOUN cana-1326 3 71 - - PUNCT cana-1326 3 72 metric metric ADJ cana-1326 3 73 space space NOUN cana-1326 3 74 . . PUNCT cana-1326 4 1 our our PRON cana-1326 4 2 results result NOUN cana-1326 4 3 generalised generalise VERB cana-1326 4 4 well well ADV cana-1326 4 5 known know VERB cana-1326 4 6 results result NOUN cana-1326 4 7 in in ADP cana-1326 4 8 the the DET cana-1326 4 9 literature literature NOUN cana-1326 4 10 . . PUNCT cana-1326 5 1 keywords keyword NOUN cana-1326 5 2 : : PUNCT cana-1326 5 3 common common ADJ cana-1326 5 4 fixed fix VERB cana-1326 5 5 point point NOUN cana-1326 5 6 , , PUNCT cana-1326 5 7 bicomplex bicomplex NOUN cana-1326 5 8 valued value VERB cana-1326 5 9 metric metric ADJ cana-1326 5 10 space space NOUN cana-1326 5 11 . . PUNCT cana-1326 6 1 2010 2010 NUM cana-1326 7 1 msc:47h09;47h10;30g35;46n9;54h25 msc:47h09;47h10;30g35;46n9;54h25 NOUN cana-1326 7 2 . . PROPN cana-1326 8 1 1 1 X cana-1326 8 2 . . X cana-1326 8 3 introduction introduction NOUN cana-1326 8 4 the the DET cana-1326 8 5 theory theory NOUN cana-1326 8 6 of of ADP cana-1326 8 7 bicomplex bicomplex NOUN cana-1326 8 8 numbers number NOUN cana-1326 8 9 have have AUX cana-1326 8 10 been be AUX cana-1326 8 11 studied study VERB cana-1326 8 12 for for ADP cana-1326 8 13 quite quite DET cana-1326 8 14 a a DET cana-1326 8 15 long long ADJ cana-1326 8 16 time time NOUN cana-1326 8 17 , , PUNCT cana-1326 8 18 which which PRON cana-1326 8 19 probably probably ADV cana-1326 8 20 began begin VERB cana-1326 8 21 with with ADP cana-1326 8 22 the the DET cana-1326 8 23 works work NOUN cana-1326 8 24 [ [ X cana-1326 8 25 3,4,5].the 3,4,5].the DET cana-1326 8 26 algebra algebra NOUN cana-1326 8 27 of of ADP cana-1326 8 28 bicomplex bicomplex NOUN cana-1326 8 29 numbers number NOUN cana-1326 8 30 are be AUX cana-1326 8 31 widely widely ADV cana-1326 8 32 used use VERB cana-1326 8 33 in in ADP cana-1326 8 34 the the DET cana-1326 8 35 literature literature NOUN cana-1326 8 36 as as SCONJ cana-1326 8 37 it it PRON cana-1326 8 38 becomes become VERB cana-1326 8 39 a a DET cana-1326 8 40 viable viable ADJ cana-1326 8 41 commutative commutative ADJ cana-1326 8 42 alternative alternative NOUN cana-1326 9 1 [ [ X cana-1326 9 2 4,5 4,5 NUM cana-1326 9 3 ] ] PUNCT cana-1326 9 4 to to ADP cana-1326 9 5 the the DET cana-1326 9 6 non non ADJ cana-1326 9 7 commutative commutative ADJ cana-1326 9 8 skew skew ADJ cana-1326 9 9 field field NOUN cana-1326 9 10 of of ADP cana-1326 9 11 quaternions quaternion NOUN cana-1326 9 12 ( ( PUNCT cana-1326 9 13 both both PRON cana-1326 9 14 are be AUX cana-1326 9 15 four four NUM cana-1326 9 16 - - PUNCT cana-1326 9 17 dimensional dimensional ADJ cana-1326 9 18 and and CCONJ cana-1326 9 19 generalization generalization NOUN cana-1326 9 20 of of ADP cana-1326 9 21 complex complex ADJ cana-1326 9 22 numbers).the numbers).the DET cana-1326 9 23 commutativity commutativity NOUN cana-1326 9 24 in in ADP cana-1326 9 25 the the DET cana-1326 9 26 former former ADJ cana-1326 9 27 is be AUX cana-1326 9 28 gained gain VERB cana-1326 9 29 at at ADP cana-1326 9 30 the the DET cana-1326 9 31 cost cost NOUN cana-1326 9 32 of of ADP cana-1326 9 33 the the DET cana-1326 9 34 fact fact NOUN cana-1326 9 35 that that SCONJ cana-1326 9 36 the the DET cana-1326 9 37 ring ring NOUN cana-1326 9 38 of of ADP cana-1326 9 39 these these DET cana-1326 9 40 numbers number NOUN cana-1326 9 41 contains contain VERB cana-1326 9 42 zero zero NUM cana-1326 9 43 - - PUNCT cana-1326 9 44 divisors divisor NOUN cana-1326 9 45 and and CCONJ cana-1326 9 46 so so ADV cana-1326 9 47 can can AUX cana-1326 9 48 not not PART cana-1326 9 49 form form VERB cana-1326 9 50 a a DET cana-1326 9 51 field field NOUN cana-1326 9 52 .it .it PUNCT cana-1326 9 53 is be AUX cana-1326 9 54 well well ADV cana-1326 9 55 known know VERB cana-1326 9 56 that that SCONJ cana-1326 9 57 the the DET cana-1326 9 58 fixed fix VERB cana-1326 9 59 point point NOUN cana-1326 9 60 theory theory NOUN cana-1326 9 61 plays play VERB cana-1326 9 62 a a DET cana-1326 9 63 very very ADV cana-1326 9 64 important important ADJ cana-1326 9 65 role role NOUN cana-1326 9 66 in in ADP cana-1326 9 67 theory theory NOUN cana-1326 9 68 and and CCONJ cana-1326 9 69 applications application NOUN cana-1326 9 70 , , PUNCT cana-1326 9 71 in in ADP cana-1326 9 72 particular particular ADJ cana-1326 9 73 , , PUNCT cana-1326 9 74 whose whose DET cana-1326 9 75 importance importance NOUN cana-1326 9 76 comes come VERB cana-1326 9 77 from from ADP cana-1326 9 78 finding find VERB cana-1326 9 79 roots root NOUN cana-1326 9 80 of of ADP cana-1326 9 81 algebraic algebraic ADJ cana-1326 9 82 equation equation NOUN cana-1326 9 83 and and CCONJ cana-1326 9 84 numerical numerical ADJ cana-1326 9 85 analysis analysis NOUN cana-1326 9 86 . . PUNCT cana-1326 10 1 banach banach NOUN cana-1326 10 2 contraction contraction NOUN cana-1326 10 3 principle principle NOUN cana-1326 10 4 in in ADP cana-1326 10 5 [ [ X cana-1326 10 6 15 15 NUM cana-1326 10 7 ] ] PUNCT cana-1326 10 8 gives give VERB cana-1326 10 9 appropriate appropriate ADJ cana-1326 10 10 and and CCONJ cana-1326 10 11 simple simple ADJ cana-1326 10 12 conditions condition NOUN cana-1326 10 13 to to PART cana-1326 10 14 establish establish VERB cana-1326 10 15 the the DET cana-1326 10 16 existence existence NOUN cana-1326 10 17 and and CCONJ cana-1326 10 18 uniqueness uniqueness NOUN cana-1326 10 19 of of ADP cana-1326 10 20 a a DET cana-1326 10 21 solution solution NOUN cana-1326 10 22 of of ADP cana-1326 10 23 an an DET cana-1326 10 24 operator operator NOUN cana-1326 10 25 equation equation NOUN cana-1326 10 26 𝑇 𝑇 PROPN cana-1326 10 27 οΏ½ οΏ½ PROPN cana-1326 10 28 π‘₯ π‘₯ NOUN cana-1326 10 29 οΏ½ οΏ½ PROPN cana-1326 10 30 = = SYM cana-1326 10 31 π‘₯ π‘₯ NOUN cana-1326 10 32 οΏ½ οΏ½ PROPN cana-1326 10 33 . . PUNCT cana-1326 11 1 later later ADV cana-1326 11 2 , , PUNCT cana-1326 11 3 a a DET cana-1326 11 4 number number NOUN cana-1326 11 5 of of ADP cana-1326 11 6 papers paper NOUN cana-1326 11 7 were be AUX cana-1326 11 8 devoted devote VERB cana-1326 11 9 to to ADP cana-1326 11 10 the the DET cana-1326 11 11 improvement improvement NOUN cana-1326 11 12 and and CCONJ cana-1326 11 13 generalization generalization NOUN cana-1326 11 14 of of ADP cana-1326 11 15 that that DET cana-1326 11 16 result result NOUN cana-1326 11 17 . . PUNCT cana-1326 12 1 most most ADJ cana-1326 12 2 of of ADP cana-1326 12 3 these these DET cana-1326 12 4 results result NOUN cana-1326 12 5 deal deal VERB cana-1326 12 6 with with ADP cana-1326 12 7 the the DET cana-1326 12 8 generalizations generalization NOUN cana-1326 12 9 of of ADP cana-1326 12 10 the the DET cana-1326 12 11 different different ADJ cana-1326 12 12 contractive contractive ADJ cana-1326 12 13 conditions condition NOUN cana-1326 12 14 in in ADP cana-1326 12 15 metric metric ADJ cana-1326 12 16 spaces space NOUN cana-1326 12 17 [ [ X cana-1326 12 18 7,10,11,17 7,10,11,17 X cana-1326 12 19 ] ] PUNCT cana-1326 12 20 . . PUNCT cana-1326 13 1 there there PRON cana-1326 13 2 have have AUX cana-1326 13 3 been be AUX cana-1326 13 4 a a DET cana-1326 13 5 number number NOUN cana-1326 13 6 of of ADP cana-1326 13 7 generalizations generalization NOUN cana-1326 13 8 of of ADP cana-1326 13 9 metric metric ADJ cana-1326 13 10 spaces space NOUN cana-1326 13 11 such such ADJ cana-1326 13 12 as as ADP cana-1326 13 13 vector vector NOUN cana-1326 13 14 valued value VERB cana-1326 13 15 metric metric ADJ cana-1326 13 16 spaces space NOUN cana-1326 13 17 , , PUNCT cana-1326 13 18 𝐺 𝐺 PROPN cana-1326 13 19 οΏ½ οΏ½ PROPN cana-1326 13 20 metric metric ADJ cana-1326 13 21 spaces space NOUN cana-1326 13 22 , , PUNCT cana-1326 13 23 pseudometric pseudometric ADJ cana-1326 13 24 spaces space NOUN cana-1326 13 25 , , PUNCT cana-1326 13 26 fuzzy fuzzy ADJ cana-1326 13 27 metric metric ADJ cana-1326 13 28 spaces space NOUN cana-1326 13 29 , , PUNCT cana-1326 13 30 𝐷 𝐷 PROPN cana-1326 13 31 οΏ½ οΏ½ NOUN cana-1326 13 32 -metric -metric ADJ cana-1326 13 33 spaces space NOUN cana-1326 13 34 , , PUNCT cana-1326 13 35 cone cone NOUN cana-1326 13 36 metric metric ADJ cana-1326 13 37 spaces space NOUN cana-1326 13 38 , , PUNCT cana-1326 13 39 and and CCONJ cana-1326 13 40 modular modular ADJ cana-1326 13 41 metric metric ADJ cana-1326 13 42 spaces space NOUN cana-1326 13 43 . . PUNCT cana-1326 14 1 bakhtin bakhtin NOUN cana-1326 15 1 [ [ X cana-1326 15 2 14 14 NUM cana-1326 15 3 ] ] PUNCT cana-1326 15 4 introduced introduce VERB cana-1326 15 5 the the DET cana-1326 15 6 notion notion NOUN cana-1326 15 7 of of ADP cana-1326 15 8 𝑏 𝑏 DET cana-1326 15 9 οΏ½ οΏ½ NOUN cana-1326 15 10 -metric -metric ADJ cana-1326 15 11 space space NOUN cana-1326 15 12 which which PRON cana-1326 15 13 is be AUX cana-1326 15 14 a a DET cana-1326 15 15 generalized generalized ADJ cana-1326 15 16 form form NOUN cana-1326 15 17 of of ADP cana-1326 15 18 metric metric ADJ cana-1326 15 19 spaces space NOUN cana-1326 15 20 . . PUNCT cana-1326 16 1 azam azam PROPN cana-1326 16 2 et et PROPN cana-1326 16 3 al al PROPN cana-1326 16 4 . . PUNCT cana-1326 17 1 [ [ X cana-1326 17 2 2,9 2,9 NUM cana-1326 17 3 ] ] PUNCT cana-1326 17 4 introduced introduce VERB cana-1326 17 5 the the DET cana-1326 17 6 notion notion NOUN cana-1326 17 7 of of ADP cana-1326 17 8 complex complex NOUN cana-1326 17 9 - - PUNCT cana-1326 17 10 valued value VERB cana-1326 17 11 metric metric ADJ cana-1326 17 12 space space NOUN cana-1326 17 13 which which PRON cana-1326 17 14 is be AUX cana-1326 17 15 a a DET cana-1326 17 16 generalization generalization NOUN cana-1326 17 17 of of ADP cana-1326 17 18 classical classical ADJ cana-1326 17 19 metric metric ADJ cana-1326 17 20 space space NOUN cana-1326 17 21 and and CCONJ cana-1326 17 22 established establish VERB cana-1326 17 23 sufficient sufficient ADJ cana-1326 17 24 conditions condition NOUN cana-1326 17 25 for for ADP cana-1326 17 26 the the DET cana-1326 17 27 existence existence NOUN cana-1326 17 28 of of ADP cana-1326 17 29 common common ADJ cana-1326 17 30 fixed fix VERB cana-1326 17 31 points point NOUN cana-1326 17 32 of of ADP cana-1326 17 33 a a DET cana-1326 17 34 pair pair NOUN cana-1326 17 35 of of ADP cana-1326 17 36 mappings mapping NOUN cana-1326 17 37 satisfying satisfy VERB cana-1326 17 38 a a DET cana-1326 17 39 contractive contractive ADJ cana-1326 17 40 condition condition NOUN cana-1326 17 41 . . PUNCT cana-1326 18 1 the the DET cana-1326 18 2 concept concept NOUN cana-1326 18 3 of of ADP cana-1326 18 4 complex complex ADJ cana-1326 18 5 valued value VERB cana-1326 18 6 𝑏 𝑏 PRON cana-1326 18 7 οΏ½ οΏ½ NOUN cana-1326 18 8 -metric -metric ADJ cana-1326 18 9 spaces space NOUN cana-1326 18 10 was be AUX cana-1326 18 11 introduced introduce VERB cana-1326 18 12 in in ADP cana-1326 18 13 2013 2013 NUM cana-1326 18 14 by by ADP cana-1326 18 15 rao rao PROPN cana-1326 18 16 et et PROPN cana-1326 18 17 al al PROPN cana-1326 18 18 . . PUNCT cana-1326 19 1 [ [ X cana-1326 19 2 18 18 NUM cana-1326 19 3 ] ] PUNCT cana-1326 19 4 . . PUNCT cana-1326 20 1 in in ADP cana-1326 20 2 sequel sequel NOUN cana-1326 20 3 , , PUNCT cana-1326 20 4 mukheimer mukheimer PROPN cana-1326 20 5 [ [ X cana-1326 20 6 16 16 NUM cana-1326 20 7 ] ] PUNCT cana-1326 20 8 proved prove VERB cana-1326 20 9 some some DET cana-1326 20 10 common common ADJ cana-1326 20 11 fixed fix VERB cana-1326 20 12 point point NOUN cana-1326 20 13 theorems theorem NOUN cana-1326 20 14 in in ADP cana-1326 20 15 complex complex NOUN cana-1326 20 16 valued value VERB cana-1326 20 17 𝑏 𝑏 PRON cana-1326 20 18 οΏ½ οΏ½ NOUN cana-1326 20 19 -metric -metric ADJ cana-1326 20 20 spaces space NOUN cana-1326 20 21 . . PUNCT cana-1326 21 1 recently recently ADV cana-1326 21 2 junesang junesang PROPN cana-1326 21 3 choi choi PROPN cana-1326 21 4 et et PROPN cana-1326 21 5 al al PROPN cana-1326 21 6 . . PUNCT cana-1326 22 1 [ [ PUNCT cana-1326 22 2 1 1 X cana-1326 22 3 ] ] PUNCT cana-1326 22 4 introduced introduce VERB cana-1326 22 5 the the DET cana-1326 22 6 notion notion NOUN cana-1326 22 7 of of ADP cana-1326 22 8 bi bi ADJ cana-1326 22 9 - - ADJ cana-1326 22 10 complex complex ADJ cana-1326 22 11 valued value VERB cana-1326 22 12 metric metric ADJ cana-1326 22 13 space space NOUN cana-1326 22 14 which which PRON cana-1326 22 15 is be AUX cana-1326 22 16 a a DET cana-1326 22 17 generalization generalization NOUN cana-1326 22 18 of of ADP cana-1326 22 19 classical classical ADJ cana-1326 22 20 metric metric ADJ cana-1326 22 21 space space NOUN cana-1326 22 22 and and CCONJ cana-1326 22 23 proved prove VERB cana-1326 22 24 certain certain ADJ cana-1326 22 25 common common ADJ cana-1326 22 26 fixed fix VERB cana-1326 22 27 point point NOUN cana-1326 22 28 theorems theorem NOUN cana-1326 22 29 for for ADP cana-1326 22 30 a a DET cana-1326 22 31 pair pair NOUN cana-1326 22 32 of of ADP cana-1326 22 33 weakly weakly ADJ cana-1326 22 34 compatible compatible ADJ cana-1326 22 35 mappings mapping NOUN cana-1326 22 36 satisfying satisfying ADJ cana-1326 22 37 ( ( PUNCT cana-1326 22 38 clrg clrg NOUN cana-1326 22 39 ) ) PUNCT cana-1326 22 40 ( ( PUNCT cana-1326 22 41 or or CCONJ cana-1326 22 42 ( ( PUNCT cana-1326 22 43 e.a e.a PROPN cana-1326 22 44 ) ) PUNCT cana-1326 22 45 ) ) PUNCT cana-1326 22 46 property property NOUN cana-1326 22 47 in in ADP cana-1326 22 48 the the DET cana-1326 22 49 bicomplex bicomplex NOUN cana-1326 22 50 valued value VERB cana-1326 22 51 metric metric ADJ cana-1326 22 52 spaces space NOUN cana-1326 22 53 . . PUNCT cana-1326 23 1 in in ADP cana-1326 23 2 2019 2019 NUM cana-1326 23 3 jebril jebril NOUN cana-1326 23 4 et.al.[8 et.al.[8 PROPN cana-1326 23 5 ] ] PUNCT cana-1326 23 6 proved prove VERB cana-1326 23 7 some some DET cana-1326 23 8 important important ADJ cana-1326 23 9 theorems theorem NOUN cana-1326 23 10 on on ADP cana-1326 23 11 common common ADJ cana-1326 23 12 fixed fix VERB cana-1326 23 13 point point NOUN cana-1326 23 14 theorems theorem NOUN cana-1326 23 15 under under ADP cana-1326 23 16 rational rational ADJ cana-1326 23 17 contractions contraction NOUN cana-1326 23 18 for for ADP cana-1326 23 19 pair pair NOUN cana-1326 23 20 of of ADP cana-1326 23 21 mappings mapping NOUN cana-1326 23 22 in in ADP cana-1326 23 23 bicomplex bicomplex NOUN cana-1326 23 24 valued value VERB cana-1326 23 25 metric metric ADJ cana-1326 23 26 spaces space NOUN cana-1326 23 27 . . PUNCT cana-1326 24 1 in in ADP cana-1326 24 2 this this DET cana-1326 24 3 communications communication NOUN cana-1326 24 4 on on ADP cana-1326 24 5 applied apply VERB cana-1326 24 6 nonlinear nonlinear ADJ cana-1326 24 7 analysis analysis NOUN cana-1326 24 8 issn issn NOUN cana-1326 24 9 : : PUNCT cana-1326 24 10 1074 1074 NUM cana-1326 24 11 - - PUNCT cana-1326 24 12 133x 133x NUM cana-1326 24 13 vol vol NOUN cana-1326 24 14 31 31 NUM cana-1326 24 15 no no NOUN cana-1326 24 16 . . PUNCT cana-1326 25 1 7s 7 NOUN cana-1326 25 2 ( ( PUNCT cana-1326 25 3 2024 2024 NUM cana-1326 25 4 ) ) PUNCT cana-1326 25 5 467 467 NUM cana-1326 25 6 https://internationalpubls.com https://internationalpubls.com X cana-1326 25 7 article article NOUN cana-1326 25 8 , , PUNCT cana-1326 25 9 we we PRON cana-1326 25 10 extend extend VERB cana-1326 25 11 and and CCONJ cana-1326 25 12 generalised generalise VERB cana-1326 25 13 the the DET cana-1326 25 14 results result NOUN cana-1326 25 15 of of ADP cana-1326 25 16 ahmad ahmad PROPN cana-1326 25 17 et.al.[13 et.al.[13 PROPN cana-1326 25 18 ] ] PUNCT cana-1326 25 19 , , PUNCT cana-1326 25 20 dubey dubey PROPN cana-1326 25 21 et et PROPN cana-1326 25 22 al al PROPN cana-1326 25 23 . . PUNCT cana-1326 26 1 [ [ X cana-1326 26 2 17 17 NUM cana-1326 26 3 ] ] PUNCT cana-1326 26 4 and and CCONJ cana-1326 26 5 rao rao NOUN cana-1326 26 6 et et PROPN cana-1326 26 7 al al PROPN cana-1326 26 8 . . PUNCT cana-1326 27 1 [ [ X cana-1326 27 2 18 18 NUM cana-1326 27 3 ] ] PUNCT cana-1326 27 4 and and CCONJ cana-1326 27 5 to to PART cana-1326 27 6 establish establish VERB cana-1326 27 7 the the DET cana-1326 27 8 existence existence NOUN cana-1326 27 9 and and CCONJ cana-1326 27 10 uniqueness uniqueness NOUN cana-1326 27 11 of of ADP cana-1326 27 12 common common ADJ cana-1326 27 13 fixed fix VERB cana-1326 27 14 points point NOUN cana-1326 27 15 for for ADP cana-1326 27 16 pair pair NOUN cana-1326 27 17 of of ADP cana-1326 27 18 self self NOUN cana-1326 27 19 mappings mapping NOUN cana-1326 27 20 on on ADP cana-1326 27 21 a a DET cana-1326 27 22 closed closed ADJ cana-1326 27 23 ball ball NOUN cana-1326 27 24 in in ADP cana-1326 27 25 bicomplex bicomplex NOUN cana-1326 27 26 valued value VERB cana-1326 27 27 b b X cana-1326 27 28 - - PUNCT cana-1326 27 29 metric metric ADJ cana-1326 27 30 space space NOUN cana-1326 27 31 which which PRON cana-1326 27 32 extends extend VERB cana-1326 27 33 a a DET cana-1326 27 34 recent recent ADJ cana-1326 27 35 results result NOUN cana-1326 27 36 of of ADP cana-1326 27 37 , , PUNCT cana-1326 27 38 i.beg i.beg NOUN cana-1326 27 39 , , PUNCT cana-1326 27 40 s.k.datta s.k.datta NOUN cana-1326 27 41 and and CCONJ cana-1326 27 42 d.pal d.pal NOUN cana-1326 28 1 [ [ X cana-1326 28 2 6 6 NUM cana-1326 28 3 ] ] PUNCT cana-1326 28 4 , , PUNCT cana-1326 28 5 md md PROPN cana-1326 28 6 . . PROPN cana-1326 28 7 a.hoque a.hoque PROPN cana-1326 29 1 [ [ X cana-1326 29 2 12 12 NUM cana-1326 29 3 ] ] PUNCT cana-1326 29 4 and and CCONJ cana-1326 29 5 several several ADJ cana-1326 29 6 others.here others.here NUM cana-1326 29 7 in in ADP cana-1326 29 8 the the DET cana-1326 29 9 following following NOUN cana-1326 29 10 , , PUNCT cana-1326 29 11 the the DET cana-1326 29 12 set set NOUN cana-1326 29 13 of of ADP cana-1326 29 14 bicomplex bicomplex NOUN cana-1326 29 15 numbers number NOUN cana-1326 29 16 , , PUNCT cana-1326 29 17 complex complex ADJ cana-1326 29 18 numbers number NOUN cana-1326 29 19 and and CCONJ cana-1326 29 20 real real ADJ cana-1326 29 21 numbers number NOUN cana-1326 29 22 are be AUX cana-1326 29 23 denoted denote VERB cana-1326 29 24 by by ADP cana-1326 29 25 β„‚2 β„‚2 NOUN cana-1326 29 26 , , PUNCT cana-1326 29 27 β„‚ β„‚ PROPN cana-1326 29 28 and and CCONJ cana-1326 29 29 β„‚0 β„‚0 NOUN cana-1326 29 30 respectively respectively ADV cana-1326 29 31 . . PUNCT cana-1326 30 1 β„‚2 β„‚2 PROPN cana-1326 30 2 becomes become VERB cana-1326 30 3 a a DET cana-1326 30 4 real real ADV cana-1326 30 5 commutative commutative ADJ cana-1326 30 6 algebra algebra NOUN cana-1326 30 7 with with ADP cana-1326 30 8 the the DET cana-1326 30 9 identity identity NOUN cana-1326 30 10 1=1+𝑖1 1=1+𝑖1 NUM cana-1326 30 11 . . PUNCT cana-1326 31 1 0+𝑖2.0+𝑖1𝑖2 0+𝑖2.0+𝑖1𝑖2 NUM cana-1326 31 2 οΏ½ οΏ½ PROPN cana-1326 31 3 .0 .0 NUM cana-1326 31 4 , , PUNCT cana-1326 31 5 the the DET cana-1326 31 6 set set NOUN cana-1326 31 7 of of ADP cana-1326 31 8 bicomplex bicomplex NOUN cana-1326 31 9 number number NOUN cana-1326 31 10 is be AUX cana-1326 31 11 defined define VERB cana-1326 31 12 as as ADP cana-1326 31 13 β„‚2={ΞΎ=π‘Ž0+π‘Ž1𝑖1+𝑖2. β„‚2={ΞΎ=π‘Ž0+π‘Ž1𝑖1+𝑖2. NOUN cana-1326 31 14 οΏ½ οΏ½ NOUN cana-1326 31 15 π‘Ž2+𝑖1𝑖2 π‘Ž2+𝑖1𝑖2 NOUN cana-1326 31 16 οΏ½ οΏ½ PROPN cana-1326 31 17 . . PUNCT cana-1326 31 18 οΏ½ οΏ½ PROPN cana-1326 31 19 π‘Ž3: π‘Ž3: PROPN cana-1326 31 20 οΏ½ οΏ½ PROPN cana-1326 31 21 π‘Ž0,π‘Ž1, π‘Ž0,π‘Ž1, PROPN cana-1326 31 22 οΏ½ οΏ½ NOUN cana-1326 31 23 π‘Ž2 π‘Ž2 NOUN cana-1326 31 24 , , PUNCT cana-1326 31 25 π‘Ž3 π‘Ž3 PROPN cana-1326 31 26 ∈ ∈ PROPN cana-1326 31 27 β„‚0π‘Žπ‘›π‘‘ β„‚0π‘Žπ‘›π‘‘ PROPN cana-1326 31 28 οΏ½ οΏ½ PROPN cana-1326 31 29 𝑖1 𝑖1 PROPN cana-1326 31 30 2 2 NUM cana-1326 31 31 = = SYM cana-1326 31 32 𝑖2 𝑖2 PROPN cana-1326 31 33 2 2 NUM cana-1326 31 34 = = SYM cana-1326 31 35 βˆ’1 βˆ’1 NOUN cana-1326 31 36 } } PUNCT cana-1326 31 37 . . PUNCT cana-1326 32 1 definition definition NOUN cana-1326 32 2 1 1 NUM cana-1326 32 3 : : PUNCT cana-1326 32 4 let let VERB cana-1326 32 5 ΞΎ1=𝑒1+𝑖2 ΞΎ1=𝑒1+𝑖2 PROPN cana-1326 32 6 οΏ½ οΏ½ PROPN cana-1326 32 7 𝑒2 𝑒2 PROPN cana-1326 32 8 ∈ ∈ PROPN cana-1326 32 9 β„‚2 β„‚2 NOUN cana-1326 32 10 and and CCONJ cana-1326 32 11 ΞΎ2=𝑣1+𝑖2 ΞΎ2=𝑣1+𝑖2 PROPN cana-1326 32 12 οΏ½ οΏ½ PROPN cana-1326 32 13 𝑣2 𝑣2 PROPN cana-1326 32 14 ∈ ∈ PROPN cana-1326 32 15 β„‚2 β„‚2 NOUN cana-1326 32 16 define define VERB cana-1326 32 17 partial partial ADJ cana-1326 32 18 order order NOUN cana-1326 32 19 relation relation NOUN cana-1326 32 20 ≲𝑖2 ≲𝑖2 PROPN cana-1326 32 21 οΏ½ οΏ½ PROPN cana-1326 32 22 οΏ½ οΏ½ PROPN cana-1326 32 23 on on ADP cana-1326 32 24 β„‚2 β„‚2 NOUN cana-1326 32 25 as as SCONJ cana-1326 32 26 follows follow VERB cana-1326 32 27 ( ( PUNCT cana-1326 32 28 see see VERB cana-1326 32 29 , , PUNCT cana-1326 32 30 e.g. e.g. ADV cana-1326 32 31 [ [ X cana-1326 32 32 6 6 NUM cana-1326 32 33 ] ] PUNCT cana-1326 32 34 ): ): PUNCT cana-1326 32 35 οΏ½ οΏ½ PROPN cana-1326 32 36 ΞΎ1 ΞΎ1 PROPN cana-1326 32 37 ≲𝑖2 ≲𝑖2 PROPN cana-1326 32 38 οΏ½ οΏ½ PROPN cana-1326 32 39 ΞΎ2 ΞΎ2 PROPN cana-1326 32 40 if if SCONJ cana-1326 32 41 and and CCONJ cana-1326 32 42 only only ADV cana-1326 32 43 if if SCONJ cana-1326 32 44 οΏ½ οΏ½ NOUN cana-1326 32 45 𝑒1 𝑒1 PROPN cana-1326 32 46 ≲ ≲ PROPN cana-1326 32 47 οΏ½ οΏ½ PROPN cana-1326 32 48 𝑣1 𝑣1 PROPN cana-1326 32 49 οΏ½ οΏ½ PROPN cana-1326 32 50 οΏ½ οΏ½ PROPN cana-1326 32 51 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ PROPN cana-1326 32 52 οΏ½ οΏ½ PROPN cana-1326 32 53 οΏ½ οΏ½ PROPN cana-1326 32 54 οΏ½ οΏ½ PROPN cana-1326 32 55 𝑒2 𝑒2 PROPN cana-1326 32 56 ≲ ≲ PROPN cana-1326 32 57 οΏ½ οΏ½ PROPN cana-1326 32 58 𝑣2 𝑣2 NOUN cana-1326 32 59 … … PUNCT cana-1326 32 60 … … PUNCT cana-1326 32 61 … … PUNCT cana-1326 32 62 .. .. PUNCT cana-1326 32 63 ( ( PUNCT cana-1326 32 64 1 1 X cana-1326 32 65 ) ) PUNCT cana-1326 32 66 where where SCONJ cana-1326 32 67 ≲ ≲ PROPN cana-1326 32 68 οΏ½ οΏ½ PROPN cana-1326 32 69 is be AUX cana-1326 32 70 the the DET cana-1326 32 71 partial partial ADJ cana-1326 32 72 order order NOUN cana-1326 32 73 on on ADP cana-1326 32 74 β„‚1 β„‚1 PROPN cana-1326 32 75 ( ( PUNCT cana-1326 32 76 see see VERB cana-1326 32 77 , , PUNCT cana-1326 32 78 e.g e.g PROPN cana-1326 33 1 [ [ X cana-1326 33 2 2 2 NUM cana-1326 33 3 ] ] PUNCT cana-1326 33 4 ) ) PUNCT cana-1326 33 5 . . PUNCT cana-1326 34 1 thus thus ADV cana-1326 34 2 ΞΎ1 ΞΎ1 VERB cana-1326 34 3 ≲𝑖2 ≲𝑖2 PROPN cana-1326 34 4 οΏ½ οΏ½ PROPN cana-1326 34 5 ΞΎ2 ΞΎ2 PROPN cana-1326 34 6 if if SCONJ cana-1326 34 7 any any DET cana-1326 34 8 one one NUM cana-1326 34 9 of of ADP cana-1326 34 10 the the DET cana-1326 34 11 following follow VERB cana-1326 34 12 properties property NOUN cana-1326 34 13 holds hold VERB cana-1326 34 14 : : PUNCT cana-1326 35 1 [ [ X cana-1326 35 2 bπ‘œ1 bπ‘œ1 X cana-1326 35 3 ] ] X cana-1326 35 4 if if SCONJ cana-1326 35 5 οΏ½ οΏ½ NOUN cana-1326 35 6 𝑒1 𝑒1 PROPN cana-1326 35 7 = = PROPN cana-1326 35 8 οΏ½ οΏ½ PROPN cana-1326 35 9 𝑣1 𝑣1 PROPN cana-1326 35 10 οΏ½ οΏ½ PROPN cana-1326 35 11 οΏ½ οΏ½ PROPN cana-1326 35 12 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ PROPN cana-1326 35 13 οΏ½ οΏ½ PROPN cana-1326 35 14 οΏ½ οΏ½ PROPN cana-1326 35 15 οΏ½ οΏ½ PROPN cana-1326 35 16 𝑒2 𝑒2 PROPN cana-1326 35 17 = = PROPN cana-1326 35 18 𝑣2 𝑣2 PROPN cana-1326 35 19 ; ; PUNCT cana-1326 35 20 [ [ X cana-1326 35 21 bπ‘œ2 bπ‘œ2 NOUN cana-1326 35 22 ] ] X cana-1326 35 23 if if SCONJ cana-1326 35 24 οΏ½ οΏ½ NOUN cana-1326 35 25 𝑒1 𝑒1 PROPN cana-1326 35 26 β‰Ί β‰Ί PROPN cana-1326 35 27 οΏ½ οΏ½ PROPN cana-1326 35 28 𝑣1 𝑣1 PROPN cana-1326 35 29 οΏ½ οΏ½ PROPN cana-1326 35 30 οΏ½ οΏ½ PROPN cana-1326 35 31 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ PROPN cana-1326 35 32 οΏ½ οΏ½ PROPN cana-1326 35 33 οΏ½ οΏ½ PROPN cana-1326 35 34 οΏ½ οΏ½ PROPN cana-1326 35 35 𝑒2 𝑒2 PROPN cana-1326 35 36 = = SYM cana-1326 35 37 οΏ½ οΏ½ PROPN cana-1326 35 38 𝑣2 𝑣2 PROPN cana-1326 35 39 ; ; PUNCT cana-1326 35 40 [ [ X cana-1326 35 41 bπ‘œ3 bπ‘œ3 X cana-1326 35 42 ] ] X cana-1326 35 43 if if SCONJ cana-1326 35 44 οΏ½ οΏ½ NOUN cana-1326 35 45 𝑒1 𝑒1 PROPN cana-1326 35 46 = = PROPN cana-1326 35 47 οΏ½ οΏ½ PROPN cana-1326 35 48 𝑣1 𝑣1 PROPN cana-1326 35 49 οΏ½ οΏ½ PROPN cana-1326 35 50 οΏ½ οΏ½ PROPN cana-1326 35 51 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ PROPN cana-1326 35 52 οΏ½ οΏ½ PROPN cana-1326 35 53 οΏ½ οΏ½ PROPN cana-1326 35 54 οΏ½ οΏ½ PROPN cana-1326 35 55 𝑒2 𝑒2 PROPN cana-1326 35 56 β‰Ί β‰Ί NOUN cana-1326 35 57 οΏ½ οΏ½ PROPN cana-1326 35 58 𝑣2 𝑣2 PROPN cana-1326 35 59 ; ; PUNCT cana-1326 35 60 [ [ X cana-1326 35 61 bπ‘œ4 bπ‘œ4 X cana-1326 35 62 ] ] X cana-1326 35 63 if if SCONJ cana-1326 35 64 𝑒1 𝑒1 PROPN cana-1326 35 65 β‰Ί β‰Ί PROPN cana-1326 35 66 οΏ½ οΏ½ PROPN cana-1326 35 67 𝑣1 𝑣1 PROPN cana-1326 35 68 οΏ½ οΏ½ PROPN cana-1326 35 69 οΏ½ οΏ½ PROPN cana-1326 35 70 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ PROPN cana-1326 35 71 οΏ½ οΏ½ PROPN cana-1326 35 72 οΏ½ οΏ½ PROPN cana-1326 35 73 οΏ½ οΏ½ PROPN cana-1326 35 74 𝑒2 𝑒2 PROPN cana-1326 35 75 β‰Ί β‰Ί PROPN cana-1326 35 76 οΏ½ οΏ½ PROPN cana-1326 35 77 𝑣2 𝑣2 NUM cana-1326 35 78 . . PUNCT cana-1326 36 1 we we PRON cana-1326 36 2 write write VERB cana-1326 36 3 ΞΎ1 ΞΎ1 PROPN cana-1326 36 4 ≰𝑖2 ≰𝑖2 PROPN cana-1326 36 5 οΏ½ οΏ½ PROPN cana-1326 36 6 ΞΎ2 ΞΎ2 PROPN cana-1326 36 7 if if SCONJ cana-1326 36 8 ΞΎ1 ΞΎ1 PROPN cana-1326 36 9 ≲𝑖2 ≲𝑖2 PROPN cana-1326 36 10 οΏ½ οΏ½ PROPN cana-1326 36 11 ΞΎ2 ΞΎ2 NOUN cana-1326 36 12 and and CCONJ cana-1326 36 13 ΞΎ1 ΞΎ1 PROPN cana-1326 36 14 β‰  β‰  PROPN cana-1326 36 15 οΏ½ οΏ½ PROPN cana-1326 36 16 ΞΎ2 ΞΎ2 NOUN cana-1326 36 17 i.e i.e X cana-1326 36 18 , , PUNCT cana-1326 36 19 one one NUM cana-1326 36 20 of of ADP cana-1326 36 21 [ [ X cana-1326 36 22 bπ‘œ2 bπ‘œ2 NOUN cana-1326 36 23 ] ] X cana-1326 36 24 , , PUNCT cana-1326 37 1 [ [ X cana-1326 37 2 bπ‘œ3 bπ‘œ3 X cana-1326 37 3 ] ] PUNCT cana-1326 37 4 and and CCONJ cana-1326 37 5 [ [ X cana-1326 37 6 bπ‘œ4 bπ‘œ4 X cana-1326 37 7 ] ] X cana-1326 37 8 is be AUX cana-1326 37 9 satisfied satisfied ADJ cana-1326 37 10 and and CCONJ cana-1326 37 11 we we PRON cana-1326 37 12 write write VERB cana-1326 37 13 ΞΎ1 ΞΎ1 PROPN cana-1326 37 14 ≺𝑖2 ≺𝑖2 PROPN cana-1326 37 15 οΏ½ οΏ½ PROPN cana-1326 37 16 ΞΎ2 ΞΎ2 PROPN cana-1326 37 17 if if SCONJ cana-1326 37 18 only only ADV cana-1326 37 19 [ [ X cana-1326 37 20 bπ‘œ4 bπ‘œ4 X cana-1326 37 21 ] ] X cana-1326 37 22 is be AUX cana-1326 37 23 satisfied satisfied ADJ cana-1326 37 24 . . PUNCT cana-1326 38 1 the the DET cana-1326 38 2 norm norm NOUN cana-1326 38 3 ||.|| ||.|| NOUN cana-1326 38 4 : : PUNCT cana-1326 38 5 β„‚2 β„‚2 X cana-1326 38 6 β†’ β†’ SYM cana-1326 38 7 β„‚0 β„‚0 PROPN cana-1326 38 8 + + CCONJ cana-1326 38 9 ( ( PUNCT cana-1326 38 10 the the DET cana-1326 38 11 set set NOUN cana-1326 38 12 of of ADP cana-1326 38 13 all all DET cana-1326 38 14 non non PRON cana-1326 38 15 negative negative ADJ cana-1326 38 16 real real ADJ cana-1326 38 17 numbers number NOUN cana-1326 38 18 ) ) PUNCT cana-1326 38 19 of of ADP cana-1326 38 20 a a DET cana-1326 38 21 bicomplex bicomplex NOUN cana-1326 38 22 number number NOUN cana-1326 38 23 is be AUX cana-1326 38 24 defined define VERB cana-1326 38 25 as as ADP cana-1326 38 26 || || PROPN cana-1326 38 27 ΞΎ||=β„‚1βˆšπ‘Ž0 ΞΎ||=β„‚1βˆšπ‘Ž0 NOUN cana-1326 38 28 2 2 NUM cana-1326 38 29 + + CCONJ cana-1326 38 30 π‘Ž1 π‘Ž1 ADP cana-1326 38 31 2 2 NUM cana-1326 39 1 + + CCONJ cana-1326 39 2 π‘Ž2 π‘Ž2 NOUN cana-1326 39 3 2 2 NUM cana-1326 39 4 + + CCONJ cana-1326 39 5 π‘Ž3 π‘Ž3 PROPN cana-1326 39 6 2 2 NUM cana-1326 39 7 … … SYM cana-1326 39 8 … … PUNCT cana-1326 39 9 … … PUNCT cana-1326 39 10 … … PUNCT cana-1326 39 11 … … PUNCT cana-1326 39 12 … … PUNCT cana-1326 39 13 … … PUNCT cana-1326 39 14 ( ( PUNCT cana-1326 39 15 2 2 NUM cana-1326 39 16 ) ) PUNCT cana-1326 39 17 for for ADP cana-1326 39 18 any any DET cana-1326 39 19 two two NUM cana-1326 39 20 bicomplex bicomplex NOUN cana-1326 39 21 numbers number NOUN cana-1326 39 22 ΞΎ1 ΞΎ1 NOUN cana-1326 39 23 , , PUNCT cana-1326 39 24 ΞΎ2 ΞΎ2 PROPN cana-1326 39 25 ∈ ∈ PROPN cana-1326 39 26 β„‚2 β„‚2 NOUN cana-1326 39 27 , , PUNCT cana-1326 39 28 one one PRON cana-1326 39 29 can can AUX cana-1326 39 30 easily easily ADV cana-1326 39 31 verify verify VERB cana-1326 39 32 that that SCONJ cana-1326 39 33 0 0 NUM cana-1326 39 34 ≲𝑖2 ≲𝑖2 ADJ cana-1326 39 35 ΞΎ1 ΞΎ1 NOUN cana-1326 39 36 ≲𝑖2 ≲𝑖2 ADJ cana-1326 39 37 οΏ½ οΏ½ PROPN cana-1326 39 38 ΞΎ2 ΞΎ2 PROPN cana-1326 39 39 β‡’ β‡’ NOUN cana-1326 39 40 ||ΞΎ1|| ||ΞΎ1|| NOUN cana-1326 39 41 ≀ ≀ NUM cana-1326 39 42 ||ΞΎ2|| ||ΞΎ2|| NUM cana-1326 39 43 ; ; PUNCT cana-1326 39 44 ||ΞΎ1 ||ΞΎ1 PROPN cana-1326 39 45 + + NOUN cana-1326 39 46 οΏ½ οΏ½ PROPN cana-1326 39 47 ΞΎ2|| ΞΎ2|| ADJ cana-1326 39 48 ≀ ≀ NOUN cana-1326 39 49 ||ΞΎ1|| ||ΞΎ1|| NOUN cana-1326 39 50 + + CCONJ cana-1326 39 51 ||ΞΎ2|| ||ΞΎ2|| NOUN cana-1326 39 52 ; ; PUNCT cana-1326 39 53 ||ΞΎ1 ||ΞΎ1 NOUN cana-1326 39 54 . . PUNCT cana-1326 40 1 ΞΎ2|| ΞΎ2|| PROPN cana-1326 40 2 ≀ ≀ NUM cana-1326 40 3 √2||ΞΎ1|| √2||ΞΎ1|| NOUN cana-1326 40 4 . . PUNCT cana-1326 41 1 ||ΞΎ2|| ||ΞΎ2|| VERB cana-1326 41 2 and and CCONJ cana-1326 41 3 ||a ||a PROPN cana-1326 41 4 ΞΎ||≀ ΞΎ||≀ PROPN cana-1326 41 5 οΏ½ οΏ½ PROPN cana-1326 41 6 a|| a|| PROPN cana-1326 41 7 οΏ½ οΏ½ PROPN cana-1326 41 8 ΞΎ|| ΞΎ|| NOUN cana-1326 41 9 where where SCONJ cana-1326 41 10 a∈ a∈ PROPN cana-1326 41 11 β„‚0 β„‚0 PROPN cana-1326 41 12 + + X cana-1326 41 13 . . PUNCT cana-1326 42 1 bicomplex bicomplex NOUN cana-1326 42 2 metric metric ADJ cana-1326 42 3 space space NOUN cana-1326 42 4 : : PUNCT cana-1326 42 5 choi choi NOUN cana-1326 42 6 et et PROPN cana-1326 42 7 al al PROPN cana-1326 42 8 . . PUNCT cana-1326 43 1 [ [ X cana-1326 43 2 1 1 X cana-1326 43 3 ] ] PUNCT cana-1326 43 4 define define VERB cana-1326 43 5 the the DET cana-1326 43 6 bicomplex bicomplex NOUN cana-1326 43 7 valued value VERB cana-1326 43 8 metric metric ADJ cana-1326 43 9 space space NOUN cana-1326 43 10 as as ADP cana-1326 43 11 : : PUNCT cana-1326 43 12 definition definition NOUN cana-1326 43 13 2 2 NUM cana-1326 43 14 : : PUNCT cana-1326 43 15 let let VERB cana-1326 43 16 x x PRON cana-1326 43 17 be be AUX cana-1326 43 18 a a DET cana-1326 43 19 non non X cana-1326 43 20 empty empty ADJ cana-1326 43 21 set set NOUN cana-1326 43 22 . . PUNCT cana-1326 44 1 suppose suppose VERB cana-1326 44 2 the the DET cana-1326 44 3 mapping mapping NOUN cana-1326 44 4 d d NOUN cana-1326 44 5 : : PUNCT cana-1326 44 6 x x PROPN cana-1326 44 7 Γ—x Γ—x ADP cana-1326 44 8 β†’ β†’ SYM cana-1326 44 9 β„‚2 β„‚2 NOUN cana-1326 44 10 satisfies satisfie NOUN cana-1326 44 11 the the DET cana-1326 44 12 following follow VERB cana-1326 44 13 conditions condition NOUN cana-1326 44 14 : : PUNCT cana-1326 45 1 [ [ X cana-1326 45 2 1 1 NUM cana-1326 45 3 ] ] PUNCT cana-1326 45 4 0 0 NUM cana-1326 46 1 ≲𝑖2 ≲𝑖2 ADJ cana-1326 46 2 d(x d(x PROPN cana-1326 46 3 , , PUNCT cana-1326 46 4 y y NOUN cana-1326 46 5 ) ) PUNCT cana-1326 46 6 for for ADP cana-1326 46 7 all all DET cana-1326 46 8 x x ADJ cana-1326 46 9 , , PUNCT cana-1326 46 10 y∈ y∈ PROPN cana-1326 46 11 𝑋 𝑋 PROPN cana-1326 46 12 ; ; PUNCT cana-1326 46 13 [ [ X cana-1326 46 14 2 2 NUM cana-1326 46 15 ] ] PUNCT cana-1326 46 16 d(x d(x NOUN cana-1326 46 17 , , PUNCT cana-1326 46 18 y)=0 y)=0 PRON cana-1326 46 19 if if SCONJ cana-1326 46 20 and and CCONJ cana-1326 46 21 only only ADV cana-1326 46 22 if if SCONJ cana-1326 46 23 x x NOUN cana-1326 46 24 = = NOUN cana-1326 46 25 y y PROPN cana-1326 46 26 ; ; PUNCT cana-1326 46 27 [ [ X cana-1326 46 28 3 3 NUM cana-1326 46 29 ] ] SYM cana-1326 46 30 d(x d(x PROPN cana-1326 46 31 , , PUNCT cana-1326 46 32 y)=d(y y)=d(y NOUN cana-1326 46 33 , , PUNCT cana-1326 46 34 x x NOUN cana-1326 46 35 ) ) PUNCT cana-1326 46 36 for for ADP cana-1326 46 37 all all DET cana-1326 46 38 x x ADJ cana-1326 46 39 , , PUNCT cana-1326 46 40 y∈ y∈ PROPN cana-1326 46 41 𝑋 𝑋 PROPN cana-1326 46 42 ; ; PUNCT cana-1326 46 43 [ [ X cana-1326 46 44 4 4 NUM cana-1326 46 45 ] ] PUNCT cana-1326 46 46 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-1326 46 47 , , PUNCT cana-1326 46 48 𝑦 𝑦 X cana-1326 46 49 ) ) PUNCT cana-1326 46 50 ≲𝑖2 ≲𝑖2 ADJ cana-1326 46 51 d(x d(x PROPN cana-1326 46 52 , , PUNCT cana-1326 46 53 z)+d(z z)+d(z X cana-1326 46 54 , , PUNCT cana-1326 46 55 y y NOUN cana-1326 46 56 ) ) PUNCT cana-1326 46 57 for for ADP cana-1326 46 58 all all DET cana-1326 46 59 x x ADJ cana-1326 46 60 , , PUNCT cana-1326 46 61 y∈ y∈ PROPN cana-1326 46 62 𝑋 𝑋 NOUN cana-1326 46 63 . . PUNCT cana-1326 47 1 then then ADV cana-1326 47 2 ( ( PUNCT cana-1326 47 3 x x X cana-1326 47 4 , , PUNCT cana-1326 47 5 d d NOUN cana-1326 47 6 ) ) PUNCT cana-1326 47 7 is be AUX cana-1326 47 8 called call VERB cana-1326 47 9 a a DET cana-1326 47 10 bicomplex bicomplex NOUN cana-1326 47 11 valued value VERB cana-1326 47 12 metric metric ADJ cana-1326 47 13 space space NOUN cana-1326 47 14 . . PUNCT cana-1326 48 1 communications communication NOUN cana-1326 48 2 on on ADP cana-1326 48 3 applied apply VERB cana-1326 48 4 nonlinear nonlinear ADJ cana-1326 48 5 analysis analysis NOUN cana-1326 48 6 issn issn NOUN cana-1326 48 7 : : PUNCT cana-1326 48 8 1074 1074 NUM cana-1326 48 9 - - PUNCT cana-1326 48 10 133x 133x NUM cana-1326 48 11 vol vol NOUN cana-1326 48 12 31 31 NUM cana-1326 48 13 no no NOUN cana-1326 48 14 . . PUNCT cana-1326 49 1 7s 7 NOUN cana-1326 49 2 ( ( PUNCT cana-1326 49 3 2024 2024 NUM cana-1326 49 4 ) ) PUNCT cana-1326 49 5 468 468 NUM cana-1326 49 6 https://internationalpubls.com https://internationalpubls.com X cana-1326 49 7 definition definition NOUN cana-1326 49 8 3:[1 3:[1 NUM cana-1326 49 9 ] ] X cana-1326 49 10 a a DET cana-1326 49 11 sequence sequence NOUN cana-1326 49 12 in in ADP cana-1326 49 13 a a DET cana-1326 49 14 nonempty nonempty ADV cana-1326 49 15 set set VERB cana-1326 49 16 x x PUNCT cana-1326 49 17 is be AUX cana-1326 49 18 a a DET cana-1326 49 19 function function NOUN cana-1326 49 20 x x NOUN cana-1326 49 21 : : PUNCT cana-1326 49 22 β„• β„• PROPN cana-1326 49 23 β†’ β†’ SYM cana-1326 49 24 β„‚2 β„‚2 PROPN cana-1326 49 25 , , PUNCT cana-1326 49 26 which which PRON cana-1326 49 27 is be AUX cana-1326 49 28 expressed express VERB cana-1326 49 29 by by ADP cana-1326 49 30 its its PRON cana-1326 49 31 range range NOUN cana-1326 49 32 set set NOUN cana-1326 49 33 { { PUNCT cana-1326 49 34 π‘₯𝑛 π‘₯𝑛 NOUN cana-1326 49 35 } } PUNCT cana-1326 49 36 where where SCONJ cana-1326 49 37 x(n)= x(n)= PROPN cana-1326 49 38 οΏ½ οΏ½ PROPN cana-1326 49 39 π‘₯𝑛 π‘₯𝑛 NOUN cana-1326 49 40 οΏ½ οΏ½ PROPN cana-1326 49 41 (n∈ (n∈ PROPN cana-1326 49 42 β„• β„• PROPN cana-1326 49 43 ) ) PUNCT cana-1326 49 44 . . PUNCT cana-1326 50 1 let let VERB cana-1326 50 2 { { PUNCT cana-1326 50 3 π‘₯𝑛 π‘₯𝑛 AUX cana-1326 50 4 } } PUNCT cana-1326 50 5 be be AUX cana-1326 50 6 a a DET cana-1326 50 7 sequence sequence NOUN cana-1326 50 8 in in ADP cana-1326 50 9 bicomplex bicomplex NOUN cana-1326 50 10 valued value VERB cana-1326 50 11 metric metric ADJ cana-1326 50 12 space space NOUN cana-1326 50 13 ( ( PUNCT cana-1326 50 14 x x X cana-1326 50 15 , , PUNCT cana-1326 50 16 d d NOUN cana-1326 50 17 ) ) PUNCT cana-1326 50 18 . . PUNCT cana-1326 51 1 the the DET cana-1326 51 2 sequence sequence NOUN cana-1326 51 3 { { PUNCT cana-1326 51 4 π‘₯𝑛 π‘₯𝑛 NOUN cana-1326 51 5 } } PUNCT cana-1326 51 6 is be AUX cana-1326 51 7 said say VERB cana-1326 51 8 to to PART cana-1326 51 9 converge converge VERB cana-1326 51 10 to to ADP cana-1326 51 11 x∈ x∈ PROPN cana-1326 51 12 𝑋if 𝑋if PROPN cana-1326 51 13 and and CCONJ cana-1326 51 14 only only ADV cana-1326 51 15 if if SCONJ cana-1326 51 16 for for ADP cana-1326 51 17 any any DET cana-1326 51 18 0 0 X cana-1326 51 19 ≺𝑖2 ≺𝑖2 PROPN cana-1326 51 20 β„° β„° PROPN cana-1326 51 21 ∈ ∈ PROPN cana-1326 51 22 β„‚2 β„‚2 NOUN cana-1326 51 23 , , PUNCT cana-1326 51 24 there there PRON cana-1326 51 25 exists exist VERB cana-1326 51 26 n∈ n∈ NOUN cana-1326 51 27 β„• β„• PROPN cana-1326 51 28 depending depend VERB cana-1326 51 29 on on ADP cana-1326 51 30 β„° β„° PRON cana-1326 51 31 such such ADJ cana-1326 51 32 that that SCONJ cana-1326 51 33 d(π‘₯𝑛,x) d(π‘₯𝑛,x) PROPN cana-1326 51 34 οΏ½ οΏ½ PROPN cana-1326 51 35 ≺𝑖2 ≺𝑖2 PROPN cana-1326 51 36 β„° β„° PROPN cana-1326 51 37 as as ADP cana-1326 51 38 n n CCONJ cana-1326 51 39 > > X cana-1326 51 40 n n CCONJ cana-1326 51 41 .. .. PUNCT cana-1326 51 42 a a DET cana-1326 51 43 sequence sequence NOUN cana-1326 51 44 { { PUNCT cana-1326 51 45 π‘₯𝑛 π‘₯𝑛 NOUN cana-1326 51 46 } } PUNCT cana-1326 51 47 in in ADP cana-1326 51 48 a a DET cana-1326 51 49 bicomplex bicomplex NOUN cana-1326 51 50 valued value VERB cana-1326 51 51 metric metric ADJ cana-1326 51 52 space space NOUN cana-1326 51 53 ( ( PUNCT cana-1326 51 54 x x X cana-1326 51 55 , , PUNCT cana-1326 51 56 d d NOUN cana-1326 51 57 ) ) PUNCT cana-1326 51 58 is be AUX cana-1326 51 59 said say VERB cana-1326 51 60 to to PART cana-1326 51 61 be be AUX cana-1326 51 62 cauchy cauchy ADJ cana-1326 51 63 sequence sequence NOUN cana-1326 51 64 if if SCONJ cana-1326 51 65 and and CCONJ cana-1326 51 66 only only ADV cana-1326 51 67 if if SCONJ cana-1326 51 68 for for ADP cana-1326 51 69 any any DET cana-1326 51 70 0 0 X cana-1326 51 71 ≺𝑖2 ≺𝑖2 PROPN cana-1326 51 72 β„° β„° PROPN cana-1326 51 73 ∈ ∈ PROPN cana-1326 51 74 β„‚2 β„‚2 NOUN cana-1326 51 75 , , PUNCT cana-1326 51 76 there there PRON cana-1326 51 77 exists exist VERB cana-1326 51 78 n n DET cana-1326 51 79 ∈ ∈ PROPN cana-1326 51 80 β„• β„• PROPN cana-1326 51 81 depending depend VERB cana-1326 51 82 on on ADP cana-1326 51 83 β„° β„° PRON cana-1326 51 84 such such ADJ cana-1326 51 85 that that SCONJ cana-1326 51 86 d(π‘₯𝑛, d(π‘₯𝑛, PROPN cana-1326 51 87 οΏ½ οΏ½ PROPN cana-1326 51 88 π‘₯π‘š) π‘₯π‘š) SYM cana-1326 51 89 οΏ½ οΏ½ NOUN cana-1326 51 90 ≺𝑖2 ≺𝑖2 PROPN cana-1326 51 91 β„° β„° PROPN cana-1326 51 92 as as ADP cana-1326 51 93 n n NUM cana-1326 51 94 , , PUNCT cana-1326 51 95 m m PROPN cana-1326 51 96 > > X cana-1326 51 97 n. n. PROPN cana-1326 51 98 a a DET cana-1326 51 99 bicomplex bicomplex NOUN cana-1326 51 100 valued value VERB cana-1326 51 101 metric metric ADJ cana-1326 51 102 space space NOUN cana-1326 51 103 is be AUX cana-1326 51 104 said say VERB cana-1326 51 105 to to PART cana-1326 51 106 be be AUX cana-1326 51 107 complete complete ADJ cana-1326 51 108 if if SCONJ cana-1326 51 109 and and CCONJ cana-1326 51 110 only only ADV cana-1326 51 111 if if SCONJ cana-1326 51 112 every every DET cana-1326 51 113 cauchy cauchy ADJ cana-1326 51 114 sequence sequence NOUN cana-1326 51 115 in in ADP cana-1326 51 116 x x PART cana-1326 51 117 converges converge NOUN cana-1326 51 118 in in ADP cana-1326 51 119 x. x. NOUN cana-1326 51 120 definition definition NOUN cana-1326 51 121 4 4 NUM cana-1326 51 122 . . PUNCT cana-1326 52 1 let let VERB cana-1326 52 2 𝑋 𝑋 PROPN cana-1326 52 3 οΏ½ οΏ½ PROPN cana-1326 52 4 be be AUX cana-1326 52 5 a a DET cana-1326 52 6 nonempty nonempty ADV cana-1326 52 7 set set VERB cana-1326 52 8 and and CCONJ cana-1326 52 9 let let VERB cana-1326 52 10 𝑠 𝑠 PART cana-1326 52 11 οΏ½ οΏ½ PROPN cana-1326 52 12 β‰₯1 β‰₯1 PROPN cana-1326 52 13 be be AUX cana-1326 52 14 a a DET cana-1326 52 15 given give VERB cana-1326 52 16 real real ADJ cana-1326 52 17 number number NOUN cana-1326 52 18 . . PUNCT cana-1326 53 1 a a DET cana-1326 53 2 function function NOUN cana-1326 53 3 d d NOUN cana-1326 53 4 : : PUNCT cana-1326 53 5 𝑋 𝑋 PROPN cana-1326 53 6 οΏ½ οΏ½ PROPN cana-1326 53 7 ×𝑋 ×𝑋 PART cana-1326 53 8 οΏ½ οΏ½ NOUN cana-1326 53 9 β†’ β†’ SYM cana-1326 53 10 β„‚2 β„‚2 NOUN cana-1326 53 11 is be AUX cana-1326 53 12 called call VERB cana-1326 53 13 a a DET cana-1326 53 14 bicomplex bicomplex NOUN cana-1326 53 15 valued value VERB cana-1326 53 16 𝑏 𝑏 DET cana-1326 53 17 οΏ½ οΏ½ NOUN cana-1326 53 18 -metric -metric NOUN cana-1326 53 19 on on ADP cana-1326 53 20 𝑋 𝑋 PROPN cana-1326 53 21 οΏ½ οΏ½ PROPN cana-1326 53 22 if if SCONJ cana-1326 53 23 for for ADP cana-1326 53 24 all all DET cana-1326 53 25 π‘₯ π‘₯ PROPN cana-1326 53 26 οΏ½ οΏ½ PROPN cana-1326 53 27 , , PUNCT cana-1326 53 28 𝑦 𝑦 NOUN cana-1326 53 29 οΏ½ οΏ½ PROPN cana-1326 53 30 , , PUNCT cana-1326 53 31 𝑧 𝑧 PROPN cana-1326 53 32 οΏ½ οΏ½ PROPN cana-1326 53 33 ∈ ∈ PROPN cana-1326 53 34 𝑋 𝑋 PROPN cana-1326 53 35 οΏ½ οΏ½ PROPN cana-1326 53 36 the the DET cana-1326 53 37 following follow VERB cana-1326 53 38 conditions condition NOUN cana-1326 53 39 are be AUX cana-1326 53 40 satisfied satisfied ADJ cana-1326 53 41 : : PUNCT cana-1326 53 42 ( ( PUNCT cana-1326 53 43 i i NOUN cana-1326 53 44 ) ) PUNCT cana-1326 53 45 0 0 PUNCT cana-1326 54 1 ≲𝑖2 ≲𝑖2 NOUN cana-1326 54 2 ( ( PUNCT cana-1326 54 3 π‘₯ π‘₯ NOUN cana-1326 54 4 οΏ½ οΏ½ PROPN cana-1326 54 5 , , PUNCT cana-1326 54 6 𝑦 𝑦 NOUN cana-1326 54 7 οΏ½ οΏ½ NOUN cana-1326 54 8 ) ) PUNCT cana-1326 54 9 and and CCONJ cana-1326 54 10 𝑑 𝑑 PROPN cana-1326 54 11 οΏ½ οΏ½ PROPN cana-1326 54 12 (π‘₯ (π‘₯ PROPN cana-1326 54 13 οΏ½ οΏ½ PROPN cana-1326 54 14 , , PUNCT cana-1326 54 15 𝑦 𝑦 NOUN cana-1326 54 16 οΏ½ οΏ½ NOUN cana-1326 54 17 ) ) PUNCT cana-1326 54 18 = = SYM cana-1326 54 19 0 0 PUNCT cana-1326 55 1 if if SCONJ cana-1326 55 2 and and CCONJ cana-1326 55 3 only only ADV cana-1326 55 4 if if SCONJ cana-1326 55 5 π‘₯ π‘₯ PROPN cana-1326 55 6 οΏ½ οΏ½ NOUN cana-1326 55 7 =𝑦 =𝑦 NOUN cana-1326 55 8 οΏ½ οΏ½ NOUN cana-1326 55 9 ; ; PUNCT cana-1326 55 10 ( ( PUNCT cana-1326 55 11 ii)d ii)d PROPN cana-1326 55 12 ( ( PUNCT cana-1326 55 13 π‘₯ π‘₯ NOUN cana-1326 55 14 οΏ½ οΏ½ PROPN cana-1326 55 15 , , PUNCT cana-1326 55 16 𝑦 𝑦 NOUN cana-1326 55 17 οΏ½ οΏ½ NOUN cana-1326 55 18 ) ) PUNCT cana-1326 55 19 = = SYM cana-1326 55 20 d(𝑦 d(𝑦 PROPN cana-1326 55 21 οΏ½ οΏ½ PROPN cana-1326 55 22 , , PUNCT cana-1326 55 23 π‘₯ π‘₯ NOUN cana-1326 55 24 οΏ½ οΏ½ PROPN cana-1326 55 25 ) ) PUNCT cana-1326 55 26 ; ; PUNCT cana-1326 55 27 ( ( PUNCT cana-1326 55 28 iii iii X cana-1326 55 29 ) ) PUNCT cana-1326 55 30 d(π‘₯ d(π‘₯ PROPN cana-1326 55 31 οΏ½ οΏ½ PROPN cana-1326 55 32 , , PUNCT cana-1326 55 33 𝑦 𝑦 NOUN cana-1326 55 34 οΏ½ οΏ½ NOUN cana-1326 55 35 ) ) PUNCT cana-1326 55 36 ≲𝑖2 ≲𝑖2 PROPN cana-1326 55 37 s[d(π‘₯ s[d(π‘₯ PROPN cana-1326 55 38 οΏ½ οΏ½ PROPN cana-1326 55 39 , , PUNCT cana-1326 55 40 𝑧 𝑧 NOUN cana-1326 55 41 οΏ½ οΏ½ NOUN cana-1326 55 42 ) ) PUNCT cana-1326 55 43 + + CCONJ cana-1326 55 44 d(𝑧 d(𝑧 NOUN cana-1326 55 45 οΏ½ οΏ½ PROPN cana-1326 55 46 , , PUNCT cana-1326 55 47 𝑦 𝑦 NOUN cana-1326 55 48 οΏ½ οΏ½ NOUN cana-1326 55 49 ) ) PUNCT cana-1326 55 50 ] ] PUNCT cana-1326 55 51 . . PUNCT cana-1326 56 1 the the DET cana-1326 56 2 pair pair NOUN cana-1326 56 3 ( ( PUNCT cana-1326 56 4 𝑋 𝑋 PROPN cana-1326 56 5 οΏ½ οΏ½ PROPN cana-1326 56 6 , , PUNCT cana-1326 56 7 𝑑 𝑑 NOUN cana-1326 56 8 οΏ½ οΏ½ NOUN cana-1326 56 9 ) ) PUNCT cana-1326 56 10 is be AUX cana-1326 56 11 called call VERB cana-1326 56 12 a a DET cana-1326 56 13 complex complex NOUN cana-1326 56 14 valued value VERB cana-1326 56 15 𝑏 𝑏 PRON cana-1326 56 16 οΏ½ οΏ½ NOUN cana-1326 56 17 -metric -metric ADJ cana-1326 56 18 space space NOUN cana-1326 56 19 . . PUNCT cana-1326 57 1 example example NOUN cana-1326 57 2 : : PUNCT cana-1326 58 1 if if SCONJ cana-1326 58 2 𝑋 𝑋 PROPN cana-1326 58 3 οΏ½ οΏ½ PROPN cana-1326 58 4 = = PUNCT cana-1326 59 1 [ [ X cana-1326 59 2 0 0 NUM cana-1326 59 3 , , PUNCT cana-1326 59 4 1 1 NUM cana-1326 59 5 ] ] PUNCT cana-1326 59 6 , , PUNCT cana-1326 59 7 define define VERB cana-1326 59 8 the the DET cana-1326 59 9 mapping mapping NOUN cana-1326 59 10 d d NOUN cana-1326 59 11 : : PUNCT cana-1326 59 12 𝑋 𝑋 PROPN cana-1326 59 13 οΏ½ οΏ½ PROPN cana-1326 59 14 ×𝑋 ×𝑋 NOUN cana-1326 59 15 οΏ½ οΏ½ NOUN cana-1326 59 16 β†’ β†’ SYM cana-1326 59 17 β„‚2 β„‚2 NOUN cana-1326 59 18 by by ADP cana-1326 59 19 d(π‘₯ d(π‘₯ PROPN cana-1326 59 20 οΏ½ οΏ½ PROPN cana-1326 59 21 , , PUNCT cana-1326 59 22 𝑦 𝑦 NOUN cana-1326 59 23 οΏ½ οΏ½ NOUN cana-1326 59 24 ) ) PUNCT cana-1326 59 25 =( =( NOUN cana-1326 59 26 1 1 NUM cana-1326 59 27 + + CCONJ cana-1326 59 28 i1 i1 PROPN cana-1326 59 29 + + CCONJ cana-1326 59 30 i2 i2 PROPN cana-1326 59 31 + + CCONJ cana-1326 59 32 i1i2)|x i1i2)|x PROPN cana-1326 59 33 οΏ½ οΏ½ PROPN cana-1326 59 34 βˆ’ βˆ’ PROPN cana-1326 59 35 οΏ½ οΏ½ PROPN cana-1326 59 36 y|2 y|2 PROPN cana-1326 59 37 , , PUNCT cana-1326 59 38 for for ADP cana-1326 59 39 all all DET cana-1326 59 40 π‘₯ π‘₯ PROPN cana-1326 59 41 οΏ½ οΏ½ PROPN cana-1326 59 42 , , PUNCT cana-1326 59 43 𝑦 𝑦 NOUN cana-1326 59 44 οΏ½ οΏ½ PROPN cana-1326 59 45 ∈ ∈ PROPN cana-1326 59 46 𝑋 𝑋 PROPN cana-1326 59 47 οΏ½ οΏ½ PROPN cana-1326 59 48 . . PUNCT cana-1326 60 1 then then ADV cana-1326 60 2 ( ( PUNCT cana-1326 60 3 𝑋 𝑋 PROPN cana-1326 60 4 οΏ½ οΏ½ PROPN cana-1326 60 5 , , PUNCT cana-1326 60 6 𝑑 𝑑 NOUN cana-1326 60 7 οΏ½ οΏ½ NOUN cana-1326 60 8 ) ) PUNCT cana-1326 60 9 is be AUX cana-1326 60 10 bicomplex bicomplex NOUN cana-1326 60 11 valued value VERB cana-1326 60 12 𝑏 𝑏 DET cana-1326 60 13 οΏ½ οΏ½ PROPN cana-1326 60 14 metric metric ADJ cana-1326 60 15 space space NOUN cana-1326 60 16 with with ADP cana-1326 60 17 𝑠 𝑠 PROPN cana-1326 60 18 οΏ½ οΏ½ NOUN cana-1326 60 19 =2 =2 ADJ cana-1326 60 20 . . PUNCT cana-1326 61 1 2 2 X cana-1326 61 2 . . X cana-1326 61 3 main main ADJ cana-1326 61 4 results result NOUN cana-1326 61 5 theorem theorem VERB cana-1326 61 6 1 1 NUM cana-1326 61 7 : : PUNCT cana-1326 61 8 let let VERB cana-1326 61 9 ( ( PUNCT cana-1326 61 10 x x X cana-1326 61 11 , , PUNCT cana-1326 61 12 d d NOUN cana-1326 61 13 ) ) PUNCT cana-1326 61 14 be be AUX cana-1326 61 15 a a DET cana-1326 61 16 complete complete ADJ cana-1326 61 17 bicomplex bicomplex NOUN cana-1326 61 18 valued value VERB cana-1326 61 19 metric metric ADJ cana-1326 61 20 space space NOUN cana-1326 61 21 with with ADP cana-1326 61 22 coefficient coefficient NOUN cana-1326 61 23 sβ‰₯ sβ‰₯ PROPN cana-1326 61 24 1 1 NUM cana-1326 61 25 and and CCONJ cana-1326 61 26 π‘₯0 π‘₯0 PROPN cana-1326 61 27 ∈ ∈ PROPN cana-1326 61 28 𝑋. 𝑋. PROPN cana-1326 61 29 0 0 NUM cana-1326 61 30 < < X cana-1326 61 31 π‘Ÿ π‘Ÿ DET cana-1326 61 32 ∈ ∈ PROPN cana-1326 61 33 β„‚ β„‚ PROPN cana-1326 61 34 and and CCONJ cana-1326 61 35 a a DET cana-1326 61 36 , , PUNCT cana-1326 61 37 b b NOUN cana-1326 61 38 , , PUNCT cana-1326 61 39 c c NOUN cana-1326 61 40 , , PUNCT cana-1326 61 41 d d NOUN cana-1326 61 42 and and CCONJ cana-1326 61 43 e e NOUN cana-1326 61 44 are be AUX cana-1326 61 45 non non ADJ cana-1326 61 46 negative negative ADJ cana-1326 61 47 reals real NOUN cana-1326 61 48 such such ADJ cana-1326 61 49 that that SCONJ cana-1326 61 50 a+√2𝐡 a+√2𝐡 PROPN cana-1326 61 51 + + CCONJ cana-1326 61 52 √2𝐢 √2𝐢 PRON cana-1326 61 53 + + NOUN cana-1326 61 54 √2𝑠𝐷 √2𝑠𝐷 NOUN cana-1326 61 55 + + CCONJ cana-1326 61 56 √2𝑠𝐸 √2𝑠𝐸 PROPN cana-1326 61 57 < < X cana-1326 61 58 1 1 X cana-1326 61 59 . . PUNCT cana-1326 62 1 let let VERB cana-1326 62 2 s s NOUN cana-1326 62 3 , , PUNCT cana-1326 62 4 t t PROPN cana-1326 62 5 : : PUNCT cana-1326 62 6 xβ†’x xβ†’x NUM cana-1326 62 7 are be AUX cana-1326 62 8 mapping map VERB cana-1326 62 9 satisfying satisfy VERB cana-1326 62 10 d(sx d(sx NOUN cana-1326 62 11 , , PUNCT cana-1326 62 12 ty ty NUM cana-1326 62 13 ) ) PUNCT cana-1326 62 14 ≲𝑖2ad(x ≲𝑖2ad(x ADV cana-1326 62 15 , , PUNCT cana-1326 62 16 y)+b y)+b PROPN cana-1326 62 17 𝑑(π‘₯,𝑆π‘₯)𝑑(𝑦,𝑇𝑦 𝑑(π‘₯,𝑆π‘₯)𝑑(𝑦,𝑇𝑦 NUM cana-1326 62 18 ) ) PUNCT cana-1326 62 19 1+𝑑(π‘₯,𝑦 1+𝑑(π‘₯,𝑦 NUM cana-1326 62 20 ) ) PUNCT cana-1326 63 1 + + CCONJ cana-1326 63 2 𝐢 𝐢 PROPN cana-1326 63 3 𝑑(𝑦,𝑆π‘₯)𝑑(π‘₯,𝑇𝑦 𝑑(𝑦,𝑆π‘₯)𝑑(π‘₯,𝑇𝑦 NOUN cana-1326 63 4 ) ) PUNCT cana-1326 63 5 1+𝑑(π‘₯,𝑦 1+𝑑(π‘₯,𝑦 NOUN cana-1326 63 6 ) ) PUNCT cana-1326 64 1 + + NUM cana-1326 64 2 𝐷 𝐷 PROPN cana-1326 64 3 𝑑(π‘₯,𝑆π‘₯)𝑑(π‘₯,𝑇𝑦 𝑑(π‘₯,𝑆π‘₯)𝑑(π‘₯,𝑇𝑦 NOUN cana-1326 64 4 ) ) PUNCT cana-1326 64 5 1+𝑑(π‘₯,𝑦 1+𝑑(π‘₯,𝑦 NOUN cana-1326 64 6 ) ) PUNCT cana-1326 65 1 + + CCONJ cana-1326 65 2 𝐸 𝐸 PROPN cana-1326 65 3 𝑑(𝑦,𝑆π‘₯)𝑑(𝑦,𝑇𝑦 𝑑(𝑦,𝑆π‘₯)𝑑(𝑦,𝑇𝑦 NOUN cana-1326 65 4 ) ) PUNCT cana-1326 65 5 1+𝑑(π‘₯,𝑦 1+𝑑(π‘₯,𝑦 NOUN cana-1326 65 6 ) ) PUNCT cana-1326 65 7 ( ( PUNCT cana-1326 65 8 1.1 1.1 NUM cana-1326 65 9 ) ) PUNCT cana-1326 65 10 for for ADP cana-1326 65 11 all all DET cana-1326 65 12 x x ADJ cana-1326 65 13 , , PUNCT cana-1326 65 14 y∈ y∈ PROPN cana-1326 65 15 b(π‘₯0 b(π‘₯0 NOUN cana-1326 65 16 , , PUNCT cana-1326 65 17 π‘Ÿ)Μ…Μ… π‘Ÿ)Μ…Μ… PROPN cana-1326 65 18 Μ…Μ… Μ…Μ… PROPN cana-1326 65 19 Μ…Μ… Μ…Μ… PROPN cana-1326 65 20 Μ…Μ… Μ…Μ… PROPN cana-1326 65 21 Μ…Μ… Μ…Μ… PROPN cana-1326 65 22 . . PUNCT cana-1326 66 1 if if SCONJ cana-1326 66 2 ‖𝑑(π‘₯0 ‖𝑑(π‘₯0 PROPN cana-1326 66 3 , , PUNCT cana-1326 66 4 𝑆π‘₯0β€– 𝑆π‘₯0β€– PRON cana-1326 66 5 ≲𝑖2 ≲𝑖2 NOUN cana-1326 66 6 ( ( PUNCT cana-1326 66 7 1 1 NUM cana-1326 66 8 βˆ’ βˆ’ PROPN cana-1326 66 9 πœ†)|π‘Ÿ| πœ†)|π‘Ÿ| PROPN cana-1326 66 10 where where SCONJ cana-1326 66 11 Ξ»= Ξ»= VERB cana-1326 66 12 max max PROPN cana-1326 66 13 { { PUNCT cana-1326 66 14 𝐴+√2𝑠𝐷 𝐴+√2𝑠𝐷 PROPN cana-1326 66 15 1βˆ’π΅βˆ’βˆš2𝑠𝐷 1βˆ’π΅βˆ’βˆš2𝑠𝐷 NUM cana-1326 66 16 , , PUNCT cana-1326 66 17 𝐴+√2𝑠𝐸 𝐴+√2𝑠𝐸 PROPN cana-1326 66 18 1βˆ’βˆš2π΅βˆ’βˆš2𝑠𝐸 1βˆ’βˆš2π΅βˆ’βˆš2𝑠𝐸 NUM cana-1326 66 19 + + ADJ cana-1326 66 20 , , PUNCT cana-1326 66 21 ( ( PUNCT cana-1326 66 22 1.2 1.2 NUM cana-1326 66 23 ) ) PUNCT cana-1326 66 24 then then ADV cana-1326 66 25 there there PRON cana-1326 66 26 exist exist VERB cana-1326 66 27 a a DET cana-1326 66 28 unique unique ADJ cana-1326 66 29 point point NOUN cana-1326 66 30 u∈ u∈ NOUN cana-1326 66 31 b(π‘₯0 b(π‘₯0 NOUN cana-1326 66 32 , , PUNCT cana-1326 66 33 π‘Ÿ)Μ…Μ… π‘Ÿ)Μ…Μ… PROPN cana-1326 66 34 Μ…Μ… Μ…Μ… PROPN cana-1326 66 35 Μ…Μ… Μ…Μ… PROPN cana-1326 66 36 Μ…Μ… Μ…Μ… PROPN cana-1326 66 37 Μ…Μ… Μ…Μ… PROPN cana-1326 66 38 such such ADJ cana-1326 66 39 that that SCONJ cana-1326 66 40 u u PROPN cana-1326 66 41 = = PROPN cana-1326 66 42 su su PROPN cana-1326 66 43 = = PROPN cana-1326 66 44 tu tu PROPN cana-1326 66 45 . . PUNCT cana-1326 66 46 proof proof NOUN cana-1326 66 47 : : PUNCT cana-1326 66 48 let let VERB cana-1326 66 49 π‘₯0 π‘₯0 PROPN cana-1326 66 50 οΏ½ οΏ½ PROPN cana-1326 66 51 be be AUX cana-1326 66 52 an an DET cana-1326 66 53 arbitrary arbitrary ADJ cana-1326 66 54 point point NOUN cana-1326 66 55 in in ADP cana-1326 66 56 x x PUNCT cana-1326 66 57 and and CCONJ cana-1326 66 58 define define VERB cana-1326 66 59 π‘₯2𝑛+1 π‘₯2𝑛+1 DET cana-1326 66 60 = = SYM cana-1326 66 61 𝑆π‘₯2𝑛 𝑆π‘₯2𝑛 PROPN cana-1326 66 62 and and CCONJ cana-1326 66 63 π‘₯2𝑛+2 π‘₯2𝑛+2 NOUN cana-1326 66 64 = = NOUN cana-1326 66 65 𝑇π‘₯2𝑛+1 𝑇π‘₯2𝑛+1 NOUN cana-1326 66 66 where where SCONJ cana-1326 66 67 n=0,1,2 n=0,1,2 NUM cana-1326 66 68 … … PUNCT cana-1326 66 69 . . PUNCT cana-1326 67 1 we we PRON cana-1326 67 2 will will AUX cana-1326 67 3 prove prove VERB cana-1326 67 4 that that SCONJ cana-1326 67 5 π‘₯𝑛 π‘₯𝑛 PROPN cana-1326 67 6 ∈ ∈ PROPN cana-1326 67 7 b(π‘₯0 b(π‘₯0 NOUN cana-1326 67 8 , , PUNCT cana-1326 67 9 π‘Ÿ)Μ…Μ… π‘Ÿ)Μ…Μ… PROPN cana-1326 67 10 Μ…Μ… Μ…Μ… PROPN cana-1326 67 11 Μ…Μ… Μ…Μ… PROPN cana-1326 67 12 Μ…Μ… Μ…Μ… PROPN cana-1326 67 13 Μ…Μ… Μ…Μ… PROPN cana-1326 67 14 for for ADP cana-1326 67 15 all all DET cana-1326 67 16 nβˆˆβ„• nβˆˆβ„• NUM cana-1326 67 17 by by ADP cana-1326 67 18 mathematical mathematical ADJ cana-1326 67 19 induction induction NOUN cana-1326 67 20 . . PUNCT cana-1326 68 1 using use VERB cana-1326 68 2 inequality inequality NOUN cana-1326 68 3 ( ( PUNCT cana-1326 68 4 1.2 1.2 NUM cana-1326 68 5 ) ) PUNCT cana-1326 68 6 and and CCONJ cana-1326 68 7 the the DET cana-1326 68 8 fact fact NOUN cana-1326 68 9 that that SCONJ cana-1326 68 10 Ξ»= Ξ»= VERB cana-1326 68 11 max max PROPN cana-1326 68 12 { { PUNCT cana-1326 68 13 𝐴+√2𝑠𝐷 𝐴+√2𝑠𝐷 PROPN cana-1326 68 14 1βˆ’π΅βˆ’βˆš2𝑠𝐷 1βˆ’π΅βˆ’βˆš2𝑠𝐷 NUM cana-1326 68 15 , , PUNCT cana-1326 68 16 𝐴+√2𝑠𝐸 𝐴+√2𝑠𝐸 PROPN cana-1326 68 17 1βˆ’βˆš2π΅βˆ’βˆš2𝑠𝐸 1βˆ’βˆš2π΅βˆ’βˆš2𝑠𝐸 NUM cana-1326 69 1 + + CCONJ cana-1326 69 2 < < X cana-1326 69 3 1 1 NUM cana-1326 69 4 we we PRON cana-1326 69 5 have have VERB cana-1326 69 6 ‖𝑑(π‘₯0 ‖𝑑(π‘₯0 PROPN cana-1326 69 7 , , PUNCT cana-1326 69 8 𝑆π‘₯0β€– 𝑆π‘₯0β€– DET cana-1326 69 9 ≲𝑖2 ≲𝑖2 PROPN cana-1326 69 10 |π‘Ÿ| |π‘Ÿ| PROPN cana-1326 69 11 οΏ½ οΏ½ PROPN cana-1326 69 12 . . PUNCT cana-1326 69 13 οΏ½ οΏ½ PROPN cana-1326 69 14 it it PRON cana-1326 69 15 implies imply VERB cana-1326 69 16 that that SCONJ cana-1326 69 17 π‘₯1 π‘₯1 PROPN cana-1326 69 18 ∈ ∈ PROPN cana-1326 69 19 b(π‘₯0 b(π‘₯0 VERB cana-1326 69 20 , , PUNCT cana-1326 69 21 π‘Ÿ)Μ…Μ… π‘Ÿ)Μ…Μ… PROPN cana-1326 69 22 Μ…Μ… Μ…Μ… PROPN cana-1326 69 23 Μ…Μ… Μ…Μ… PROPN cana-1326 69 24 Μ…Μ… Μ…Μ… PROPN cana-1326 69 25 Μ…Μ… Μ…Μ… PROPN cana-1326 69 26 . . PUNCT cana-1326 70 1 let let VERB cana-1326 70 2 π‘₯2 π‘₯2 NOUN cana-1326 70 3 , , PUNCT cana-1326 70 4 π‘₯3 π‘₯3 NOUN cana-1326 70 5 , , PUNCT cana-1326 70 6 … … PUNCT cana-1326 70 7 , , PUNCT cana-1326 70 8 π‘₯π‘˜ π‘₯π‘˜ X cana-1326 70 9 ∈ ∈ PROPN cana-1326 70 10 b(π‘₯0 b(π‘₯0 VERB cana-1326 70 11 , , PUNCT cana-1326 70 12 π‘Ÿ)Μ…Μ… π‘Ÿ)Μ…Μ… PROPN cana-1326 70 13 Μ…Μ… Μ…Μ… PROPN cana-1326 70 14 Μ…Μ… Μ…Μ… PROPN cana-1326 70 15 Μ…Μ… Μ…Μ… PROPN cana-1326 70 16 Μ…Μ… Μ…Μ… PROPN cana-1326 70 17 οΏ½ οΏ½ PROPN cana-1326 70 18 οΏ½ οΏ½ PROPN cana-1326 70 19 for for ADP cana-1326 70 20 some some DET cana-1326 70 21 kΡ”β„•. kΡ”β„•. PROPN cana-1326 70 22 if if SCONJ cana-1326 70 23 k=2n+1 k=2n+1 PROPN cana-1326 70 24 where where SCONJ cana-1326 70 25 n=0,1,2 n=0,1,2 NUM cana-1326 70 26 … … SYM cana-1326 70 27 , , PUNCT cana-1326 70 28 π‘˜βˆ’1 π‘˜βˆ’1 PROPN cana-1326 70 29 2 2 NUM cana-1326 70 30 or or CCONJ cana-1326 70 31 k=2n+2 k=2n+2 PROPN cana-1326 70 32 where where SCONJ cana-1326 70 33 n=0,1,2 n=0,1,2 NUM cana-1326 70 34 , , PUNCT cana-1326 70 35 … … PUNCT cana-1326 70 36 . . PROPN cana-1326 70 37 , , PUNCT cana-1326 70 38 π‘˜βˆ’2 π‘˜βˆ’2 PROPN cana-1326 70 39 2 2 NUM cana-1326 70 40 , , PUNCT cana-1326 70 41 we we PRON cana-1326 70 42 obtain obtain VERB cana-1326 70 43 by by ADP cana-1326 70 44 using use VERB cana-1326 70 45 inequality inequality NOUN cana-1326 70 46 ( ( PUNCT cana-1326 70 47 1.1 1.1 NUM cana-1326 70 48 ) ) PUNCT cana-1326 70 49 d(x2n+1 d(x2n+1 PROPN cana-1326 70 50 , , PUNCT cana-1326 70 51 x2n+2 x2n+2 PUNCT cana-1326 70 52 ) ) PUNCT cana-1326 71 1 = = SYM cana-1326 71 2 d(sx2n d(sx2n NOUN cana-1326 71 3 , , PUNCT cana-1326 71 4 tx2n+1 tx2n+1 NOUN cana-1326 71 5 ) ) PUNCT cana-1326 71 6 ≲𝑖2 ≲𝑖2 PROPN cana-1326 71 7 𝐴 𝐴 PROPN cana-1326 71 8 οΏ½ οΏ½ PROPN cana-1326 71 9 d(x2n d(x2n PROPN cana-1326 71 10 , , PUNCT cana-1326 71 11 x2n+1 x2n+1 PUNCT cana-1326 71 12 ) ) PUNCT cana-1326 72 1 + + CCONJ cana-1326 72 2 𝐡 𝐡 NOUN cana-1326 72 3 d(x2n+1,tx2n+1)d(x2n d(x2n+1,tx2n+1)d(x2n NOUN cana-1326 72 4 , , PUNCT cana-1326 72 5 sx2n sx2n NOUN cana-1326 72 6 ) ) PUNCT cana-1326 72 7 1+d(x2n 1+d(x2n NUM cana-1326 72 8 , , PUNCT cana-1326 72 9 x2n+1 x2n+1 PUNCT cana-1326 72 10 ) ) PUNCT cana-1326 73 1 + + ADP cana-1326 73 2 c c PROPN cana-1326 73 3 d(x2n+1,sx2n)d(x2n d(x2n+1,sx2n)d(x2n PROPN cana-1326 73 4 , , PUNCT cana-1326 73 5 tx2n+1 tx2n+1 NOUN cana-1326 73 6 ) ) PUNCT cana-1326 73 7 1+d(x2n 1+d(x2n NUM cana-1326 73 8 , , PUNCT cana-1326 73 9 x2n+1 x2n+1 NUM cana-1326 73 10 ) ) PUNCT cana-1326 73 11 communications communication NOUN cana-1326 73 12 on on ADP cana-1326 73 13 applied apply VERB cana-1326 73 14 nonlinear nonlinear ADJ cana-1326 73 15 analysis analysis NOUN cana-1326 73 16 issn issn NOUN cana-1326 73 17 : : PUNCT cana-1326 73 18 1074 1074 NUM cana-1326 73 19 - - PUNCT cana-1326 73 20 133x 133x NUM cana-1326 73 21 vol vol NOUN cana-1326 73 22 31 31 NUM cana-1326 73 23 no no NOUN cana-1326 73 24 . . PUNCT cana-1326 74 1 7s 7 NOUN cana-1326 74 2 ( ( PUNCT cana-1326 74 3 2024 2024 NUM cana-1326 74 4 ) ) PUNCT cana-1326 74 5 469 469 NUM cana-1326 74 6 https://internationalpubls.com https://internationalpubls.com X cana-1326 75 1 + + PUNCT cana-1326 75 2 d d PROPN cana-1326 75 3 d(x2n d(x2n PROPN cana-1326 75 4 , , PUNCT cana-1326 75 5 tx2n+1)d(x2n tx2n+1)d(x2n NOUN cana-1326 75 6 , , PUNCT cana-1326 75 7 sx2n sx2n NOUN cana-1326 75 8 ) ) PUNCT cana-1326 75 9 1+d(x2n 1+d(x2n NUM cana-1326 75 10 , , PUNCT cana-1326 75 11 x2n+1 x2n+1 PUNCT cana-1326 75 12 ) ) PUNCT cana-1326 76 1 + + CCONJ cana-1326 76 2 𝐸 𝐸 PROPN cana-1326 76 3 d(x2n+1,tx2n+1)d(x2n+1,sx2n d(x2n+1,tx2n+1)d(x2n+1,sx2n PROPN cana-1326 76 4 ) ) PUNCT cana-1326 76 5 1+d(x2n 1+d(x2n NUM cana-1326 76 6 , , PUNCT cana-1326 76 7 x2n+1 x2n+1 PUNCT cana-1326 76 8 ) ) PUNCT cana-1326 76 9 ≲𝑖2 ≲𝑖2 PROPN cana-1326 76 10 𝐴 𝐴 PROPN cana-1326 76 11 οΏ½ οΏ½ PROPN cana-1326 76 12 d(x2n d(x2n PROPN cana-1326 76 13 , , PUNCT cana-1326 76 14 x2n+1 x2n+1 PUNCT cana-1326 76 15 ) ) PUNCT cana-1326 77 1 + + CCONJ cana-1326 77 2 𝐡 𝐡 NOUN cana-1326 77 3 d(x2n+1,x2n+2)d(x2n d(x2n+1,x2n+2)d(x2n PROPN cana-1326 77 4 , , PUNCT cana-1326 77 5 x2n+1 x2n+1 PROPN cana-1326 77 6 ) ) PUNCT cana-1326 77 7 1+d(x2n 1+d(x2n NUM cana-1326 77 8 , , PUNCT cana-1326 77 9 x2n+1 x2n+1 PROPN cana-1326 77 10 ) ) PUNCT cana-1326 78 1 + + NOUN cana-1326 78 2 d d NOUN cana-1326 78 3 d(x2n d(x2n PROPN cana-1326 78 4 , , PUNCT cana-1326 78 5 x2n+2)d(x2n x2n+2)d(x2n NUM cana-1326 78 6 , , PUNCT cana-1326 78 7 x2n+1 x2n+1 PROPN cana-1326 78 8 ) ) PUNCT cana-1326 78 9 1+d(x2n 1+d(x2n NUM cana-1326 78 10 , , PUNCT cana-1326 78 11 x2n+1 x2n+1 PROPN cana-1326 78 12 ) ) PUNCT cana-1326 79 1 this this PRON cana-1326 79 2 implies imply VERB cana-1326 79 3 β€–d(x2n+1 β€–d(x2n+1 PROPN cana-1326 79 4 , , PUNCT cana-1326 79 5 x2n+2)β€– x2n+2)β€– PUNCT cana-1326 79 6 ≀ ≀ NUM cana-1326 79 7 𝐴‖d(x2n 𝐴‖d(x2n PROPN cana-1326 79 8 , , PUNCT cana-1326 79 9 x2n+1)β€– x2n+1)β€– PROPN cana-1326 80 1 + + NOUN cana-1326 80 2 √2𝐡 √2𝐡 PROPN cana-1326 80 3 β€–d(x2n+1,x2n+2)β€–β€–d(x2n β€–d(x2n+1,x2n+2)β€–β€–d(x2n NOUN cana-1326 80 4 , , PUNCT cana-1326 80 5 x2n+1)β€– x2n+1)β€– PROPN cana-1326 80 6 β€–1+d(x2n β€–1+d(x2n PROPN cana-1326 80 7 , , PUNCT cana-1326 80 8 x2n+1)β€– x2n+1)β€– PROPN cana-1326 80 9 + + CCONJ cana-1326 80 10 √2𝐷 √2𝐷 NUM cana-1326 80 11 β€–d(x2n β€–d(x2n NOUN cana-1326 80 12 , , PUNCT cana-1326 80 13 x2n+2)β€–β€–d(x2n x2n+2)β€–β€–d(x2n PROPN cana-1326 80 14 , , PUNCT cana-1326 80 15 x2n+1)β€– x2n+1)β€– PROPN cana-1326 80 16 β€–1+d(x2n β€–1+d(x2n PROPN cana-1326 80 17 , , PUNCT cana-1326 80 18 x2n+1)β€– x2n+1)β€– PROPN cana-1326 80 19 since since SCONJ cana-1326 80 20 β€–1 β€–1 PROPN cana-1326 80 21 + + X cana-1326 80 22 d(x2n d(x2n PROPN cana-1326 80 23 , , PUNCT cana-1326 80 24 x2n+1)β€– x2n+1)β€– PROPN cana-1326 80 25 > > X cana-1326 81 1 β€–d(x2n β€–d(x2n PROPN cana-1326 81 2 , , PUNCT cana-1326 81 3 x2n+1)β€– x2n+1)β€– PROPN cana-1326 81 4 hence hence ADV cana-1326 81 5 β€–d(x2n+1 β€–d(x2n+1 PROPN cana-1326 81 6 , , PUNCT cana-1326 81 7 x2n+2)β€– x2n+2)β€– PUNCT cana-1326 81 8 ≀ ≀ NUM cana-1326 81 9 𝐴‖d(x2n 𝐴‖d(x2n PROPN cana-1326 81 10 , , PUNCT cana-1326 81 11 x2n+1)β€– x2n+1)β€– PROPN cana-1326 81 12 + + CCONJ cana-1326 81 13 √2𝐡‖d(x2n+1 √2𝐡‖d(x2n+1 ADJ cana-1326 81 14 , , PUNCT cana-1326 81 15 x2n+2)β€– x2n+2)β€– X cana-1326 81 16 + + NUM cana-1326 81 17 √2𝐷‖d(x2n √2𝐷‖d(x2n PROPN cana-1326 81 18 , , PUNCT cana-1326 81 19 x2n+2)β€– x2n+2)β€– PUNCT cana-1326 81 20 ≀ ≀ NUM cana-1326 81 21 𝐴‖d(x2n 𝐴‖d(x2n PROPN cana-1326 81 22 , , PUNCT cana-1326 81 23 x2n+1)β€– x2n+1)β€– PROPN cana-1326 81 24 + + CCONJ cana-1326 81 25 √2𝐡‖d(x2n+1 √2𝐡‖d(x2n+1 ADJ cana-1326 81 26 , , PUNCT cana-1326 81 27 x2n+2)β€– x2n+2)β€– PROPN cana-1326 81 28 + + PROPN cana-1326 81 29 οΏ½ οΏ½ PROPN cana-1326 81 30 √2𝑠𝐷*β€–d(x2n √2𝑠𝐷*β€–d(x2n PROPN cana-1326 81 31 , , PUNCT cana-1326 81 32 x2n+1)β€– x2n+1)β€– PROPN cana-1326 81 33 + + PROPN cana-1326 81 34 β€–d(x2n+1 β€–d(x2n+1 PROPN cana-1326 81 35 , , PUNCT cana-1326 81 36 x2n+2)β€– x2n+2)β€– PROPN cana-1326 81 37 ( ( PUNCT cana-1326 81 38 1-√2𝐡 1-√2𝐡 NOUN cana-1326 81 39 βˆ’ βˆ’ PROPN cana-1326 81 40 οΏ½ οΏ½ NOUN cana-1326 81 41 √2𝑠𝐷)β€–d(x2n+1 √2𝑠𝐷)β€–d(x2n+1 NUM cana-1326 81 42 , , PUNCT cana-1326 81 43 x2n+2)β€– x2n+2)β€– X cana-1326 81 44 ≀ ≀ PROPN cana-1326 81 45 ( ( PUNCT cana-1326 81 46 𝐴 𝐴 PROPN cana-1326 81 47 + + CCONJ cana-1326 81 48 √2𝑠𝐷) √2𝑠𝐷) PROPN cana-1326 81 49 οΏ½ οΏ½ NOUN cana-1326 81 50 β€–d(x2n β€–d(x2n NOUN cana-1326 81 51 , , PUNCT cana-1326 81 52 x2n+1)β€– x2n+1)β€– PROPN cana-1326 81 53 β‡’β€–d(x2n+1 β‡’β€–d(x2n+1 PROPN cana-1326 81 54 , , PUNCT cana-1326 81 55 x2n+2)β€– x2n+2)β€– ADP cana-1326 81 56 ≀ ≀ NUM cana-1326 81 57 𝐴+√2𝑠𝐷 𝐴+√2𝑠𝐷 PROPN cana-1326 81 58 1βˆ’βˆš2π΅βˆ’βˆš2𝑠𝐷 1βˆ’βˆš2π΅βˆ’βˆš2𝑠𝐷 PROPN cana-1326 81 59 οΏ½ οΏ½ NOUN cana-1326 81 60 β€–d(x2n β€–d(x2n PROPN cana-1326 81 61 , , PUNCT cana-1326 81 62 x2n+1)β€– x2n+1)β€– PROPN cana-1326 81 63 ( ( PUNCT cana-1326 81 64 1.3 1.3 NUM cana-1326 81 65 ) ) PUNCT cana-1326 81 66 similarly similarly ADV cana-1326 81 67 we we PRON cana-1326 81 68 get get VERB cana-1326 81 69 , , PUNCT cana-1326 81 70 β€–d(x2n+2 β€–d(x2n+2 PROPN cana-1326 81 71 , , PUNCT cana-1326 81 72 x2n+3)β€– x2n+3)β€– PROPN cana-1326 81 73 ≀ ≀ NUM cana-1326 81 74 𝐴+√2𝑠𝐸 𝐴+√2𝑠𝐸 PROPN cana-1326 81 75 1βˆ’βˆš2π΅βˆ’βˆš2𝑠𝐸 1βˆ’βˆš2π΅βˆ’βˆš2𝑠𝐸 NUM cana-1326 81 76 οΏ½ οΏ½ PROPN cana-1326 81 77 β€–d(x2n+1 β€–d(x2n+1 PROPN cana-1326 81 78 , , PUNCT cana-1326 81 79 x2n+2)β€– x2n+2)β€– PROPN cana-1326 81 80 ( ( PUNCT cana-1326 81 81 1.4 1.4 NUM cana-1326 81 82 ) ) PUNCT cana-1326 81 83 putting put VERB cana-1326 81 84 Ξ»= Ξ»= ADJ cana-1326 81 85 max max PROPN cana-1326 81 86 { { PUNCT cana-1326 81 87 a+√2sd a+√2sd ADP cana-1326 81 88 1βˆ’bβˆ’βˆš2sd 1βˆ’bβˆ’βˆš2sd NUM cana-1326 81 89 , , PUNCT cana-1326 81 90 a+√2se a+√2se PROPN cana-1326 81 91 1βˆ’βˆš2bβˆ’βˆš2se 1βˆ’βˆš2bβˆ’βˆš2se NUM cana-1326 81 92 + + NUM cana-1326 81 93 , , PUNCT cana-1326 81 94 we we PRON cana-1326 81 95 οΏ½ οΏ½ VERB cana-1326 81 96 obtain obtain VERB cana-1326 81 97 οΏ½ οΏ½ PROPN cana-1326 81 98 β€–d(xk β€–d(xk X cana-1326 81 99 , , PUNCT cana-1326 81 100 xk+1)β€– xk+1)β€– NOUN cana-1326 81 101 ≀ ≀ NOUN cana-1326 81 102 πœ†π‘˜β€–π‘‘(π‘₯0 πœ†π‘˜β€–π‘‘(π‘₯0 NUM cana-1326 81 103 , , PUNCT cana-1326 81 104 π‘₯1β€– π‘₯1β€– X cana-1326 81 105 ( ( PUNCT cana-1326 81 106 1.5 1.5 NUM cana-1326 81 107 ) ) PUNCT cana-1326 81 108 for for ADP cana-1326 81 109 all all DET cana-1326 81 110 kΡ”β„• kΡ”β„• NOUN cana-1326 81 111 β€–d(x0 β€–d(x0 NOUN cana-1326 81 112 , , PUNCT cana-1326 81 113 xk+1)β€– xk+1)β€– PROPN cana-1326 81 114 ≀ ≀ PROPN cana-1326 81 115 𝑠‖𝑑(π‘₯0 𝑠‖𝑑(π‘₯0 NUM cana-1326 81 116 , , PUNCT cana-1326 81 117 π‘₯1β€– π‘₯1β€– PROPN cana-1326 81 118 + + SYM cana-1326 81 119 𝑠‖d(x1 𝑠‖d(x1 PROPN cana-1326 81 120 , , PUNCT cana-1326 81 121 xk+1)β€– xk+1)β€– NOUN cana-1326 81 122 ≀ ≀ PROPN cana-1326 81 123 𝑠‖𝑑(π‘₯0 𝑠‖𝑑(π‘₯0 NUM cana-1326 81 124 , , PUNCT cana-1326 81 125 π‘₯1β€– π‘₯1β€– X cana-1326 81 126 + + SYM cana-1326 81 127 𝑠2‖𝑑(π‘₯1 𝑠2‖𝑑(π‘₯1 ADJ cana-1326 81 128 , , PUNCT cana-1326 81 129 π‘₯2β€– π‘₯2β€– PROPN cana-1326 81 130 + + CCONJ cana-1326 81 131 𝑠2‖𝑑(π‘₯2 𝑠2‖𝑑(π‘₯2 NUM cana-1326 81 132 , , PUNCT cana-1326 81 133 π‘₯π‘˜+1β€– π‘₯π‘˜+1β€– PROPN cana-1326 81 134 ≀ ≀ PROPN cana-1326 81 135 𝑠‖𝑑(π‘₯0 𝑠‖𝑑(π‘₯0 NUM cana-1326 81 136 , , PUNCT cana-1326 81 137 π‘₯1β€– π‘₯1β€– X cana-1326 81 138 + + SYM cana-1326 81 139 𝑠2‖𝑑(π‘₯1 𝑠2‖𝑑(π‘₯1 ADJ cana-1326 81 140 , , PUNCT cana-1326 81 141 π‘₯2β€– π‘₯2β€– PROPN cana-1326 81 142 + + CCONJ cana-1326 81 143 𝑠3‖𝑑(π‘₯2 𝑠3‖𝑑(π‘₯2 NOUN cana-1326 81 144 , , PUNCT cana-1326 81 145 π‘₯3β€– π‘₯3β€– NOUN cana-1326 81 146 + + ADJ cana-1326 81 147 β‹―+ β‹―+ NOUN cana-1326 81 148 π‘ π‘˜+1‖𝑑(π‘₯π‘˜ π‘ π‘˜+1‖𝑑(π‘₯π‘˜ PROPN cana-1326 81 149 , , PUNCT cana-1326 81 150 π‘₯π‘˜+1β€– π‘₯π‘˜+1β€– PROPN cana-1326 81 151 οΏ½ οΏ½ PROPN cana-1326 81 152 ≀ ≀ NUM cana-1326 81 153 𝑠‖𝑑(π‘₯0 𝑠‖𝑑(π‘₯0 NUM cana-1326 81 154 , , PUNCT cana-1326 81 155 π‘₯1β€–+𝑠2πœ†β€–π‘‘(π‘₯0 π‘₯1β€–+𝑠2πœ†β€–π‘‘(π‘₯0 NUM cana-1326 81 156 , , PUNCT cana-1326 82 1 π‘₯1β€– π‘₯1β€– PROPN cana-1326 82 2 + + CCONJ cana-1326 82 3 𝑠3πœ†2 𝑠3πœ†2 PROPN cana-1326 82 4 ‖𝑑(π‘₯0 ‖𝑑(π‘₯0 PROPN cana-1326 82 5 , , PUNCT cana-1326 82 6 π‘₯1β€– π‘₯1β€– X cana-1326 82 7 + + NOUN cana-1326 82 8 β‹―+ β‹―+ NOUN cana-1326 83 1 π‘ π‘˜+1 π‘ π‘˜+1 PROPN cana-1326 83 2 πœ†π‘˜ πœ†π‘˜ PROPN cana-1326 83 3 ‖𝑑(π‘₯0 ‖𝑑(π‘₯0 PROPN cana-1326 83 4 , , PUNCT cana-1326 83 5 π‘₯1β€– π‘₯1β€– PROPN cana-1326 83 6 = = SYM cana-1326 83 7 ‖𝑑(π‘₯0 ‖𝑑(π‘₯0 PROPN cana-1326 83 8 , , PUNCT cana-1326 83 9 π‘₯1β€– π‘₯1β€– VERB cana-1326 84 1 [ [ X cana-1326 84 2 s+𝑠2πœ† s+𝑠2πœ† NOUN cana-1326 84 3 + + NOUN cana-1326 84 4 𝑠3πœ†2 𝑠3πœ†2 NOUN cana-1326 84 5 + + ADJ cana-1326 84 6 β‹―+ β‹―+ NOUN cana-1326 84 7 π‘ π‘˜+1 π‘ π‘˜+1 NOUN cana-1326 84 8 πœ†π‘˜ πœ†π‘˜ NUM cana-1326 84 9 ] ] PUNCT cana-1326 84 10 ≀ ≀ X cana-1326 84 11 ( ( PUNCT cana-1326 84 12 1 1 NUM cana-1326 84 13 βˆ’ βˆ’ NOUN cana-1326 84 14 πœ†)|π‘Ÿ|𝑠 πœ†)|π‘Ÿ|𝑠 NOUN cana-1326 84 15 1βˆ’(π‘ πœ†)π‘˜+1 1βˆ’(π‘ πœ†)π‘˜+1 NUM cana-1326 84 16 1βˆ’π‘ πœ† 1βˆ’π‘ πœ† NOUN cana-1326 85 1 [ [ PUNCT cana-1326 85 2 as as ADP cana-1326 85 3 ‖𝑑(π‘₯0 ‖𝑑(π‘₯0 PROPN cana-1326 85 4 , , PUNCT cana-1326 85 5 π‘₯1β€– π‘₯1β€– PROPN cana-1326 85 6 οΏ½ οΏ½ PROPN cana-1326 85 7 ≀ ≀ NOUN cana-1326 85 8 ( ( PUNCT cana-1326 85 9 1 1 NUM cana-1326 85 10 βˆ’ βˆ’ PROPN cana-1326 85 11 πœ†)|π‘Ÿ| πœ†)|π‘Ÿ| PROPN cana-1326 85 12 ≀ ≀ PROPN cana-1326 85 13 |π‘Ÿ| |π‘Ÿ| NOUN cana-1326 85 14 when when SCONJ cana-1326 85 15 s=1 s=1 PRON cana-1326 85 16 gives give VERB cana-1326 85 17 xk+1 xk+1 PROPN cana-1326 85 18 ∈b ∈b PROPN cana-1326 85 19 ( ( PUNCT cana-1326 85 20 x0 x0 PROPN cana-1326 85 21 , , PUNCT cana-1326 85 22 π‘Ÿ π‘Ÿ NOUN cana-1326 85 23 ) ) PUNCT cana-1326 85 24 . . PUNCT cana-1326 86 1 hence hence ADV cana-1326 86 2 π‘₯𝑛 π‘₯𝑛 PROPN cana-1326 86 3 ∈ ∈ PROPN cana-1326 86 4 b(π‘₯0 b(π‘₯0 NOUN cana-1326 86 5 , , PUNCT cana-1326 86 6 π‘Ÿ)Μ…Μ… π‘Ÿ)Μ…Μ… PROPN cana-1326 86 7 Μ…Μ… Μ…Μ… PROPN cana-1326 86 8 Μ…Μ… Μ…Μ… PROPN cana-1326 86 9 Μ…Μ… Μ…Μ… PROPN cana-1326 86 10 Μ…Μ… Μ…Μ… PROPN cana-1326 86 11 βˆ€π‘› βˆ€π‘› PROPN cana-1326 86 12 ∈ ∈ PROPN cana-1326 86 13 β„• β„• PROPN cana-1326 86 14 and and CCONJ cana-1326 86 15 οΏ½ οΏ½ PROPN cana-1326 86 16 β€–d(xn β€–d(xn PROPN cana-1326 86 17 , , PUNCT cana-1326 86 18 xn+1)β€– xn+1)β€– PROPN cana-1326 86 19 ≀ ≀ PROPN cana-1326 86 20 πœ†π‘›β€–π‘‘(π‘₯0 πœ†π‘›β€–π‘‘(π‘₯0 PROPN cana-1326 86 21 , , PUNCT cana-1326 86 22 π‘₯1β€– π‘₯1β€– PROPN cana-1326 86 23 βˆ€π‘› βˆ€π‘› PROPN cana-1326 86 24 ∈ ∈ PROPN cana-1326 86 25 β„• β„• PROPN cana-1326 86 26 ( ( PUNCT cana-1326 86 27 1.6 1.6 NUM cana-1326 86 28 ) ) PUNCT cana-1326 86 29 without without ADP cana-1326 86 30 loss loss NOUN cana-1326 86 31 of of ADP cana-1326 86 32 generality generality NOUN cana-1326 86 33 , , PUNCT cana-1326 86 34 we we PRON cana-1326 86 35 take take VERB cana-1326 86 36 m m PROPN cana-1326 86 37 > > X cana-1326 86 38 𝑛 𝑛 PROPN cana-1326 86 39 , , PUNCT cana-1326 86 40 then then ADV cana-1326 86 41 communications communication NOUN cana-1326 86 42 on on ADP cana-1326 86 43 applied apply VERB cana-1326 86 44 nonlinear nonlinear ADJ cana-1326 86 45 analysis analysis NOUN cana-1326 86 46 issn issn NOUN cana-1326 86 47 : : PUNCT cana-1326 86 48 1074 1074 NUM cana-1326 86 49 - - PUNCT cana-1326 86 50 133x 133x NUM cana-1326 86 51 vol vol NOUN cana-1326 86 52 31 31 NUM cana-1326 86 53 no no NOUN cana-1326 86 54 . . PUNCT cana-1326 87 1 7s 7 NOUN cana-1326 87 2 ( ( PUNCT cana-1326 87 3 2024 2024 NUM cana-1326 87 4 ) ) PUNCT cana-1326 87 5 470 470 NUM cana-1326 87 6 https://internationalpubls.com https://internationalpubls.com X cana-1326 87 7 οΏ½ οΏ½ PROPN cana-1326 87 8 β€–d(xn β€–d(xn PROPN cana-1326 87 9 , , PUNCT cana-1326 87 10 xm)β€– xm)β€– PROPN cana-1326 87 11 ≀ ≀ NUM cana-1326 87 12 𝑠‖d(xn 𝑠‖d(xn NUM cana-1326 87 13 , , PUNCT cana-1326 87 14 xn+1)β€– xn+1)β€– PROPN cana-1326 87 15 + + CCONJ cana-1326 87 16 𝑠‖d(xn+1 𝑠‖d(xn+1 PROPN cana-1326 87 17 , , PUNCT cana-1326 87 18 xm)β€– xm)β€– PROPN cana-1326 87 19 ≀ ≀ NUM cana-1326 87 20 𝑠‖d(xn 𝑠‖d(xn NUM cana-1326 87 21 , , PUNCT cana-1326 87 22 xn+1)β€– xn+1)β€– PROPN cana-1326 87 23 + + CCONJ cana-1326 87 24 𝑠2‖𝑑(π‘₯𝑛+1 𝑠2‖𝑑(π‘₯𝑛+1 ADJ cana-1326 87 25 , , PUNCT cana-1326 87 26 π‘₯𝑛+2β€– π‘₯𝑛+2β€– PROPN cana-1326 87 27 + + NOUN cana-1326 87 28 𝑠2‖𝑑(π‘₯𝑛+2 𝑠2‖𝑑(π‘₯𝑛+2 NUM cana-1326 87 29 , , PUNCT cana-1326 87 30 π‘₯π‘šβ€– π‘₯π‘šβ€– PROPN cana-1326 87 31 ≀ ≀ PROPN cana-1326 87 32 𝑠‖d(xn 𝑠‖d(xn PROPN cana-1326 87 33 , , PUNCT cana-1326 87 34 xn+1)β€– xn+1)β€– PROPN cana-1326 87 35 + + CCONJ cana-1326 87 36 𝑠2‖𝑑(π‘₯𝑛+1 𝑠2‖𝑑(π‘₯𝑛+1 ADJ cana-1326 87 37 , , PUNCT cana-1326 87 38 π‘₯𝑛+2β€– π‘₯𝑛+2β€– PRON cana-1326 87 39 + + ADJ cana-1326 87 40 β‹―+ β‹―+ NOUN cana-1326 87 41 π‘ π‘šβˆ’π‘›βˆ’1‖𝑑(π‘₯π‘šβˆ’2 π‘ π‘šβˆ’π‘›βˆ’1‖𝑑(π‘₯π‘šβˆ’2 NOUN cana-1326 87 42 , , PUNCT cana-1326 87 43 π‘₯π‘šβˆ’1β€– π‘₯π‘šβˆ’1β€– PROPN cana-1326 87 44 οΏ½ οΏ½ PROPN cana-1326 87 45 οΏ½ οΏ½ PROPN cana-1326 87 46 οΏ½ οΏ½ PROPN cana-1326 87 47 οΏ½ οΏ½ PROPN cana-1326 87 48 οΏ½ οΏ½ PROPN cana-1326 87 49 οΏ½ οΏ½ PROPN cana-1326 87 50 οΏ½ οΏ½ PROPN cana-1326 87 51 οΏ½ οΏ½ PROPN cana-1326 87 52 οΏ½ οΏ½ PROPN cana-1326 87 53 οΏ½ οΏ½ PROPN cana-1326 87 54 οΏ½ οΏ½ PROPN cana-1326 87 55 οΏ½ οΏ½ PROPN cana-1326 87 56 οΏ½ οΏ½ PROPN cana-1326 87 57 οΏ½ οΏ½ PROPN cana-1326 87 58 οΏ½ οΏ½ PROPN cana-1326 87 59 οΏ½ οΏ½ PROPN cana-1326 87 60 οΏ½ οΏ½ PROPN cana-1326 87 61 οΏ½ οΏ½ PROPN cana-1326 87 62 οΏ½ οΏ½ PROPN cana-1326 87 63 οΏ½ οΏ½ PROPN cana-1326 87 64 οΏ½ οΏ½ PROPN cana-1326 87 65 οΏ½ οΏ½ PROPN cana-1326 87 66 οΏ½ οΏ½ PROPN cana-1326 87 67 οΏ½ οΏ½ PROPN cana-1326 87 68 οΏ½ οΏ½ PROPN cana-1326 87 69 οΏ½ οΏ½ PROPN cana-1326 87 70 οΏ½ οΏ½ PROPN cana-1326 87 71 οΏ½ οΏ½ PROPN cana-1326 87 72 οΏ½ οΏ½ PROPN cana-1326 87 73 οΏ½ οΏ½ PROPN cana-1326 87 74 οΏ½ οΏ½ PROPN cana-1326 87 75 οΏ½ οΏ½ PROPN cana-1326 87 76 οΏ½ οΏ½ PROPN cana-1326 87 77 οΏ½ οΏ½ PROPN cana-1326 87 78 οΏ½ οΏ½ PROPN cana-1326 87 79 οΏ½ οΏ½ PROPN cana-1326 87 80 οΏ½ οΏ½ PROPN cana-1326 87 81 οΏ½ οΏ½ PROPN cana-1326 87 82 οΏ½ οΏ½ PROPN cana-1326 87 83 οΏ½ οΏ½ PROPN cana-1326 87 84 οΏ½ οΏ½ PROPN cana-1326 87 85 οΏ½ οΏ½ PROPN cana-1326 87 86 οΏ½ οΏ½ PROPN cana-1326 87 87 οΏ½ οΏ½ PROPN cana-1326 87 88 οΏ½ οΏ½ PROPN cana-1326 87 89 οΏ½ οΏ½ PROPN cana-1326 87 90 οΏ½ οΏ½ PROPN cana-1326 87 91 οΏ½ οΏ½ PROPN cana-1326 87 92 οΏ½ οΏ½ PROPN cana-1326 87 93 οΏ½ οΏ½ PROPN cana-1326 87 94 οΏ½ οΏ½ PROPN cana-1326 87 95 οΏ½ οΏ½ PROPN cana-1326 87 96 οΏ½ οΏ½ PROPN cana-1326 87 97 οΏ½ οΏ½ PROPN cana-1326 87 98 οΏ½ οΏ½ PROPN cana-1326 87 99 οΏ½ οΏ½ PROPN cana-1326 87 100 οΏ½ οΏ½ PROPN cana-1326 87 101 οΏ½ οΏ½ PROPN cana-1326 87 102 οΏ½ οΏ½ PROPN cana-1326 87 103 οΏ½ οΏ½ PROPN cana-1326 87 104 οΏ½ οΏ½ PROPN cana-1326 87 105 οΏ½ οΏ½ PROPN cana-1326 87 106 οΏ½ οΏ½ PROPN cana-1326 87 107 οΏ½ οΏ½ PROPN cana-1326 87 108 οΏ½ οΏ½ PROPN cana-1326 87 109 οΏ½ οΏ½ PROPN cana-1326 87 110 οΏ½ οΏ½ PROPN cana-1326 87 111 οΏ½ οΏ½ PROPN cana-1326 87 112 οΏ½ οΏ½ PROPN cana-1326 87 113 οΏ½ οΏ½ PROPN cana-1326 87 114 οΏ½ οΏ½ PROPN cana-1326 87 115 οΏ½ οΏ½ PROPN cana-1326 87 116 οΏ½ οΏ½ PROPN cana-1326 87 117 οΏ½ οΏ½ PROPN cana-1326 87 118 οΏ½ οΏ½ PROPN cana-1326 87 119 οΏ½ οΏ½ PROPN cana-1326 87 120 οΏ½ οΏ½ PROPN cana-1326 87 121 οΏ½ οΏ½ PROPN cana-1326 87 122 οΏ½ οΏ½ PROPN cana-1326 87 123 οΏ½ οΏ½ PROPN cana-1326 87 124 οΏ½ οΏ½ PROPN cana-1326 87 125 οΏ½ οΏ½ PROPN cana-1326 87 126 οΏ½ οΏ½ PROPN cana-1326 87 127 οΏ½ οΏ½ PROPN cana-1326 87 128 οΏ½ οΏ½ PROPN cana-1326 87 129 οΏ½ οΏ½ PROPN cana-1326 87 130 οΏ½ οΏ½ PROPN cana-1326 87 131 οΏ½ οΏ½ PROPN cana-1326 87 132 οΏ½ οΏ½ PROPN cana-1326 87 133 οΏ½ οΏ½ PROPN cana-1326 87 134 οΏ½ οΏ½ PROPN cana-1326 87 135 οΏ½ οΏ½ PROPN cana-1326 87 136 οΏ½ οΏ½ PROPN cana-1326 87 137 οΏ½ οΏ½ PROPN cana-1326 87 138 οΏ½ οΏ½ PROPN cana-1326 87 139 οΏ½ οΏ½ PROPN cana-1326 87 140 οΏ½ οΏ½ PROPN cana-1326 87 141 οΏ½ οΏ½ PROPN cana-1326 87 142 οΏ½ οΏ½ PROPN cana-1326 87 143 οΏ½ οΏ½ PROPN cana-1326 87 144 οΏ½ οΏ½ PROPN cana-1326 87 145 οΏ½ οΏ½ PROPN cana-1326 87 146 οΏ½ οΏ½ PROPN cana-1326 87 147 οΏ½ οΏ½ PROPN cana-1326 87 148 οΏ½ οΏ½ PROPN cana-1326 87 149 οΏ½ οΏ½ PROPN cana-1326 87 150 οΏ½ οΏ½ PROPN cana-1326 87 151 οΏ½ οΏ½ PROPN cana-1326 87 152 οΏ½ οΏ½ PROPN cana-1326 87 153 οΏ½ οΏ½ PROPN cana-1326 87 154 οΏ½ οΏ½ PROPN cana-1326 87 155 οΏ½ οΏ½ PROPN cana-1326 87 156 οΏ½ οΏ½ PROPN cana-1326 87 157 οΏ½ οΏ½ PROPN cana-1326 87 158 οΏ½ οΏ½ PROPN cana-1326 87 159 οΏ½ οΏ½ PROPN cana-1326 87 160 οΏ½ οΏ½ PROPN cana-1326 87 161 οΏ½ οΏ½ PROPN cana-1326 87 162 οΏ½ οΏ½ PROPN cana-1326 87 163 οΏ½ οΏ½ PROPN cana-1326 87 164 οΏ½ οΏ½ PROPN cana-1326 87 165 οΏ½ οΏ½ PROPN cana-1326 87 166 οΏ½ οΏ½ PROPN cana-1326 87 167 + + NOUN cana-1326 87 168 π‘ π‘šβˆ’π‘›β€–π‘‘(π‘₯π‘šβˆ’1 π‘ π‘šβˆ’π‘›β€–π‘‘(π‘₯π‘šβˆ’1 PROPN cana-1326 87 169 , , PUNCT cana-1326 87 170 π‘₯π‘šβ€– π‘₯π‘šβ€– PROPN cana-1326 87 171 by by ADP cana-1326 87 172 using use VERB cana-1326 87 173 ( ( PUNCT cana-1326 87 174 1.6 1.6 NUM cana-1326 87 175 ) ) PUNCT cana-1326 87 176 we we PRON cana-1326 87 177 get get VERB cana-1326 87 178 , , PUNCT cana-1326 87 179 οΏ½ οΏ½ PROPN cana-1326 87 180 β€–d(xn β€–d(xn PROPN cana-1326 87 181 , , PUNCT cana-1326 87 182 xm)β€– xm)β€– PROPN cana-1326 87 183 ≀ ≀ PROPN cana-1326 87 184 π‘ πœ†π‘›β€–π‘‘(π‘₯0 π‘ πœ†π‘›β€–π‘‘(π‘₯0 ADJ cana-1326 87 185 , , PUNCT cana-1326 87 186 π‘₯1β€– π‘₯1β€– PROPN cana-1326 87 187 + + NUM cana-1326 87 188 𝑠2πœ†π‘›+1‖𝑑(π‘₯0 𝑠2πœ†π‘›+1‖𝑑(π‘₯0 ADJ cana-1326 87 189 , , PUNCT cana-1326 87 190 π‘₯1β€– π‘₯1β€– PROPN cana-1326 87 191 οΏ½ οΏ½ PROPN cana-1326 87 192 οΏ½ οΏ½ PROPN cana-1326 87 193 οΏ½ οΏ½ PROPN cana-1326 87 194 οΏ½ οΏ½ PROPN cana-1326 87 195 οΏ½ οΏ½ PROPN cana-1326 87 196 οΏ½ οΏ½ PROPN cana-1326 87 197 οΏ½ οΏ½ PROPN cana-1326 87 198 οΏ½ οΏ½ PROPN cana-1326 87 199 οΏ½ οΏ½ PROPN cana-1326 87 200 οΏ½ οΏ½ PROPN cana-1326 87 201 οΏ½ οΏ½ PROPN cana-1326 87 202 οΏ½ οΏ½ PROPN cana-1326 87 203 οΏ½ οΏ½ PROPN cana-1326 87 204 οΏ½ οΏ½ PROPN cana-1326 87 205 οΏ½ οΏ½ PROPN cana-1326 87 206 οΏ½ οΏ½ PROPN cana-1326 87 207 οΏ½ οΏ½ PROPN cana-1326 87 208 οΏ½ οΏ½ PROPN cana-1326 87 209 οΏ½ οΏ½ PROPN cana-1326 87 210 οΏ½ οΏ½ PROPN cana-1326 87 211 οΏ½ οΏ½ PROPN cana-1326 87 212 οΏ½ οΏ½ PROPN cana-1326 87 213 οΏ½ οΏ½ PROPN cana-1326 87 214 οΏ½ οΏ½ PROPN cana-1326 87 215 οΏ½ οΏ½ PROPN cana-1326 87 216 οΏ½ οΏ½ PROPN cana-1326 87 217 οΏ½ οΏ½ PROPN cana-1326 87 218 οΏ½ οΏ½ PROPN cana-1326 87 219 οΏ½ οΏ½ PROPN cana-1326 87 220 οΏ½ οΏ½ PROPN cana-1326 87 221 οΏ½ οΏ½ PROPN cana-1326 87 222 οΏ½ οΏ½ PROPN cana-1326 87 223 οΏ½ οΏ½ PROPN cana-1326 87 224 οΏ½ οΏ½ PROPN cana-1326 87 225 οΏ½ οΏ½ PROPN cana-1326 87 226 οΏ½ οΏ½ PROPN cana-1326 87 227 + + NOUN cana-1326 87 228 𝑠3πœ†π‘›+2‖𝑑(π‘₯0 𝑠3πœ†π‘›+2‖𝑑(π‘₯0 ADJ cana-1326 87 229 , , PUNCT cana-1326 87 230 π‘₯1β€– π‘₯1β€– VERB cana-1326 87 231 + + ADJ cana-1326 87 232 β‹―+ β‹―+ NOUN cana-1326 87 233 οΏ½ οΏ½ NOUN cana-1326 87 234 π‘ π‘šβˆ’π‘›βˆ’1πœ†π‘šβˆ’2‖𝑑(π‘₯0 π‘ π‘šβˆ’π‘›βˆ’1πœ†π‘šβˆ’2‖𝑑(π‘₯0 VERB cana-1326 87 235 , , PUNCT cana-1326 87 236 π‘₯1β€– π‘₯1β€– PROPN cana-1326 87 237 + + NUM cana-1326 87 238 π‘ π‘šβˆ’π‘›πœ†π‘šβˆ’1‖𝑑(π‘₯0 π‘ π‘šβˆ’π‘›πœ†π‘šβˆ’1‖𝑑(π‘₯0 ADJ cana-1326 87 239 , , PUNCT cana-1326 87 240 π‘₯1β€– π‘₯1β€– PROPN cana-1326 87 241 = = PUNCT cana-1326 87 242 βˆ‘ βˆ‘ ADP cana-1326 87 243 π‘ π‘–πœ†π‘›+π‘–βˆ’1π‘šβˆ’π‘› π‘ π‘–πœ†π‘›+π‘–βˆ’1π‘šβˆ’π‘› ADJ cana-1326 87 244 𝑖=1 𝑖=1 PROPN cana-1326 87 245 ‖𝑑(π‘₯0 ‖𝑑(π‘₯0 PROPN cana-1326 87 246 , , PUNCT cana-1326 87 247 π‘₯1β€– π‘₯1β€– PROPN cana-1326 87 248 ≀ ≀ NUM cana-1326 87 249 π‘ πœ†π‘› π‘ πœ†π‘› NOUN cana-1326 87 250 1βˆ’πœ†π‘  1βˆ’πœ†π‘  NUM cana-1326 87 251 ‖𝑑(π‘₯0 ‖𝑑(π‘₯0 PROPN cana-1326 87 252 , , PUNCT cana-1326 87 253 π‘₯1β€– π‘₯1β€– PROPN cana-1326 87 254 β†’0 β†’0 PUNCT cana-1326 87 255 as as ADP cana-1326 87 256 m m PROPN cana-1326 87 257 , , PUNCT cana-1326 87 258 nβ†’βˆž nβ†’βˆž X cana-1326 87 259 this this PRON cana-1326 87 260 implies imply VERB cana-1326 87 261 that that SCONJ cana-1326 87 262 the the DET cana-1326 87 263 sequence sequence NOUN cana-1326 87 264 { { PUNCT cana-1326 87 265 π‘₯𝑛+ π‘₯𝑛+ PROPN cana-1326 87 266 οΏ½ οΏ½ PROPN cana-1326 87 267 π‘Žπ‘  π‘Žπ‘  PROPN cana-1326 87 268 οΏ½ οΏ½ PROPN cana-1326 87 269 a a DET cana-1326 87 270 cauchy cauchy ADJ cana-1326 87 271 sequence sequence NOUN cana-1326 87 272 in in ADP cana-1326 87 273 b(π‘₯0 b(π‘₯0 ADJ cana-1326 87 274 , , PUNCT cana-1326 87 275 π‘Ÿ)Μ…Μ… π‘Ÿ)Μ…Μ… PROPN cana-1326 87 276 Μ…Μ… Μ…Μ… PROPN cana-1326 87 277 Μ…Μ… Μ…Μ… PROPN cana-1326 87 278 Μ…Μ… Μ…Μ… PROPN cana-1326 87 279 Μ…Μ… Μ…Μ… PROPN cana-1326 87 280 . . PUNCT cana-1326 88 1 therefore therefore ADV cana-1326 88 2 there there PRON cana-1326 88 3 exists exist VERB cana-1326 88 4 a a DET cana-1326 88 5 point point NOUN cana-1326 88 6 u∈ u∈ PROPN cana-1326 88 7 b(π‘₯0 b(π‘₯0 NOUN cana-1326 88 8 , , PUNCT cana-1326 88 9 π‘Ÿ)Μ…Μ… π‘Ÿ)Μ…Μ… PROPN cana-1326 88 10 Μ…Μ… Μ…Μ… PROPN cana-1326 88 11 Μ…Μ… Μ…Μ… PROPN cana-1326 88 12 Μ…Μ… Μ…Μ… PROPN cana-1326 88 13 Μ…Μ… Μ…Μ… PROPN cana-1326 88 14 with with ADP cana-1326 88 15 limπ‘›β†’βˆž limπ‘›β†’βˆž PROPN cana-1326 88 16 π‘₯𝑛 π‘₯𝑛 PROPN cana-1326 88 17 = = PUNCT cana-1326 88 18 𝑒. 𝑒. NOUN cana-1326 88 19 we we PRON cana-1326 88 20 prove prove VERB cana-1326 88 21 that that SCONJ cana-1326 88 22 u u PROPN cana-1326 88 23 = = PROPN cana-1326 88 24 su su NOUN cana-1326 88 25 . . PUNCT cana-1326 89 1 let let VERB cana-1326 89 2 us we PRON cana-1326 89 3 consider consider VERB cana-1326 89 4 ‖𝑑(𝑒 ‖𝑑(𝑒 NOUN cana-1326 89 5 , , PUNCT cana-1326 89 6 𝑆𝑒)β€– 𝑆𝑒)β€– VERB cana-1326 89 7 ≀ ≀ NOUN cana-1326 89 8 𝑠‖𝑑(𝑒 𝑠‖𝑑(𝑒 NOUN cana-1326 89 9 , , PUNCT cana-1326 89 10 π‘₯2𝑛+2)β€– π‘₯2𝑛+2)β€– PROPN cana-1326 89 11 + + CCONJ cana-1326 89 12 𝑠‖𝑑(π‘₯2𝑛+2 𝑠‖𝑑(π‘₯2𝑛+2 NOUN cana-1326 89 13 , , PUNCT cana-1326 89 14 𝑆𝑒)β€– 𝑆𝑒)β€– VERB cana-1326 89 15 ≀ ≀ NOUN cana-1326 89 16 𝑠‖𝑑(𝑒 𝑠‖𝑑(𝑒 NOUN cana-1326 89 17 , , PUNCT cana-1326 89 18 π‘₯2𝑛+2)β€– π‘₯2𝑛+2)β€– PROPN cana-1326 89 19 + + CCONJ cana-1326 89 20 𝑠‖𝑑(𝑇π‘₯2𝑛+1 𝑠‖𝑑(𝑇π‘₯2𝑛+1 ADJ cana-1326 89 21 , , PUNCT cana-1326 89 22 𝑆𝑒)β€– 𝑆𝑒)β€– ADJ cana-1326 89 23 ≀ ≀ NOUN cana-1326 89 24 𝑠‖𝑑(𝑒 𝑠‖𝑑(𝑒 NOUN cana-1326 89 25 , , PUNCT cana-1326 89 26 π‘₯2𝑛+2)β€– π‘₯2𝑛+2)β€– PROPN cana-1326 89 27 + + CCONJ cana-1326 89 28 𝑠‖𝑑(𝑆𝑒 𝑠‖𝑑(𝑆𝑒 ADJ cana-1326 89 29 , , PUNCT cana-1326 89 30 𝑇π‘₯2𝑛+1)β€– 𝑇π‘₯2𝑛+1)β€– NOUN cana-1326 89 31 ≀ ≀ ADJ cana-1326 89 32 𝑠‖𝑑(𝑒 𝑠‖𝑑(𝑒 NOUN cana-1326 89 33 , , PUNCT cana-1326 89 34 π‘₯2𝑛+2)β€– π‘₯2𝑛+2)β€– PROPN cana-1326 89 35 + + CCONJ cana-1326 89 36 𝐴𝑠‖𝑑(π‘₯2𝑛+1 𝐴𝑠‖𝑑(π‘₯2𝑛+1 NOUN cana-1326 89 37 , , PUNCT cana-1326 89 38 𝑒)β€– 𝑒)β€– X cana-1326 90 1 + + CCONJ cana-1326 90 2 π‘ π΅βˆš2 π‘ π΅βˆš2 NOUN cana-1326 90 3 ‖𝑑(π‘₯2𝑛+1,𝑇π‘₯2𝑛+1)‖‖𝑑(𝑒,𝑆𝑒)β€– ‖𝑑(π‘₯2𝑛+1,𝑇π‘₯2𝑛+1)‖‖𝑑(𝑒,𝑆𝑒)β€– ADP cana-1326 90 4 β€–1+𝑑(𝑒,π‘₯2𝑛+1)β€– β€–1+𝑑(𝑒,π‘₯2𝑛+1)β€– PROPN cana-1326 90 5 οΏ½ οΏ½ PROPN cana-1326 90 6 + + NOUN cana-1326 90 7 π‘ πΆβˆš2 π‘ πΆβˆš2 NOUN cana-1326 90 8 ‖𝑑(π‘₯2𝑛+1,𝑆𝑒)‖‖𝑑(𝑒,𝑇π‘₯2𝑛+1)β€– ‖𝑑(π‘₯2𝑛+1,𝑆𝑒)‖‖𝑑(𝑒,𝑇π‘₯2𝑛+1)β€– NOUN cana-1326 91 1 β€–1+𝑑(π‘₯2𝑛+1),𝑒)β€– β€–1+𝑑(π‘₯2𝑛+1),𝑒)β€– PROPN cana-1326 91 2 + + CCONJ cana-1326 91 3 π‘ π·βˆš2 π‘ π·βˆš2 PROPN cana-1326 91 4 ‖𝑑(𝑒,𝑆𝑒)‖‖𝑑(𝑒,𝑇π‘₯2𝑛+1)β€– ‖𝑑(𝑒,𝑆𝑒)‖‖𝑑(𝑒,𝑇π‘₯2𝑛+1)β€– PROPN cana-1326 91 5 οΏ½ οΏ½ PROPN cana-1326 91 6 οΏ½ οΏ½ PROPN cana-1326 91 7 β€–1+𝑑(𝑒,π‘₯2𝑛+1)β€– β€–1+𝑑(𝑒,π‘₯2𝑛+1)β€– PROPN cana-1326 91 8 + + NUM cana-1326 91 9 π‘ πΈβˆš2 π‘ πΈβˆš2 NOUN cana-1326 91 10 ‖𝑑(π‘₯2𝑛+1,𝑇π‘₯2𝑛+1)‖‖𝑑(π‘₯2𝑛+1,𝑆𝑒)β€– ‖𝑑(π‘₯2𝑛+1,𝑇π‘₯2𝑛+1)‖‖𝑑(π‘₯2𝑛+1,𝑆𝑒)β€– NUM cana-1326 91 11 β€–1+𝑑(𝑒,π‘₯2𝑛+1)β€– β€–1+𝑑(𝑒,π‘₯2𝑛+1)β€– PROPN cana-1326 91 12 notice notice VERB cana-1326 91 13 that that SCONJ cana-1326 91 14 , , PUNCT cana-1326 91 15 limπ‘›β†’βˆžβ€–π‘‘(𝑒 limπ‘›β†’βˆžβ€–π‘‘(𝑒 ADJ cana-1326 91 16 , , PUNCT cana-1326 91 17 π‘₯2𝑛+2)β€– π‘₯2𝑛+2)β€– PROPN cana-1326 91 18 = = SYM cana-1326 91 19 limπ‘›β†’βˆžβ€–π‘‘(π‘₯2𝑛+1 limπ‘›β†’βˆžβ€–π‘‘(π‘₯2𝑛+1 PROPN cana-1326 91 20 , , PUNCT cana-1326 91 21 𝑒)β€– 𝑒)β€– X cana-1326 91 22 + + CCONJ cana-1326 91 23 lim lim NOUN cana-1326 91 24 π‘›β†’βˆž π‘›β†’βˆž NUM cana-1326 91 25 ‖𝑑(π‘₯2𝑛+1 ‖𝑑(π‘₯2𝑛+1 PROPN cana-1326 91 26 , , PUNCT cana-1326 91 27 𝑆𝑒)β€– 𝑆𝑒)β€– NOUN cana-1326 91 28 = = SYM cana-1326 91 29 0 0 NUM cana-1326 91 30 hence hence ADV cana-1326 91 31 ‖𝑑(𝑒 ‖𝑑(𝑒 NOUN cana-1326 91 32 , , PUNCT cana-1326 91 33 𝑆𝑒)β€– 𝑆𝑒)β€– ADJ cana-1326 91 34 = = NOUN cana-1326 91 35 0 0 NUM cana-1326 91 36 that that PRON cana-1326 91 37 is be AUX cana-1326 91 38 u u NOUN cana-1326 91 39 = = NOUN cana-1326 91 40 su su X cana-1326 91 41 similarly similarly ADV cana-1326 91 42 , , PUNCT cana-1326 91 43 u u NOUN cana-1326 91 44 = = PROPN cana-1326 91 45 tu tu PROPN cana-1326 91 46 for for ADP cana-1326 91 47 uniqueness uniqueness NOUN cana-1326 91 48 assume assume VERB cana-1326 91 49 that that SCONJ cana-1326 91 50 π‘’βˆ— π‘’βˆ— NOUN cana-1326 91 51 in in ADP cana-1326 91 52 b(π‘₯0 b(π‘₯0 ADJ cana-1326 91 53 , , PUNCT cana-1326 91 54 π‘Ÿ)Μ…Μ… π‘Ÿ)Μ…Μ… PROPN cana-1326 91 55 Μ…Μ… Μ…Μ… PROPN cana-1326 91 56 Μ…Μ… Μ…Μ… PROPN cana-1326 91 57 Μ…Μ… Μ…Μ… PROPN cana-1326 91 58 Μ…Μ… Μ…Μ… PROPN cana-1326 91 59 is be AUX cana-1326 91 60 a a DET cana-1326 91 61 another another DET cana-1326 91 62 common common ADJ cana-1326 91 63 fixed fix VERB cana-1326 91 64 point point NOUN cana-1326 91 65 of of ADP cana-1326 91 66 s s PRON cana-1326 91 67 and and CCONJ cana-1326 91 68 t. t. NOUN cana-1326 91 69 then then ADV cana-1326 91 70 ‖𝑑(𝑒 ‖𝑑(𝑒 NOUN cana-1326 91 71 , , PUNCT cana-1326 91 72 π‘’βˆ—)β€– π‘’βˆ—)β€– ADJ cana-1326 91 73 ≀ ≀ ADJ cana-1326 91 74 ‖𝑑(𝑆𝑒 ‖𝑑(𝑆𝑒 NOUN cana-1326 91 75 , , PUNCT cana-1326 91 76 π‘‡π‘’βˆ—)β€– π‘‡π‘’βˆ—)β€– PROPN cana-1326 91 77 ≀ ≀ PUNCT cana-1326 91 78 a‖𝑑(𝑒 a‖𝑑(𝑒 PROPN cana-1326 91 79 , , PUNCT cana-1326 91 80 π‘’βˆ—)β€– π‘’βˆ—)β€– PRON cana-1326 92 1 + + X cana-1326 92 2 𝐡√2 𝐡√2 NUM cana-1326 93 1 ‖𝑑(𝑒 ‖𝑑(𝑒 NOUN cana-1326 93 2 , , PUNCT cana-1326 93 3 𝑆𝑒)‖‖𝑑(π‘’βˆ— 𝑆𝑒)‖‖𝑑(π‘’βˆ— PROPN cana-1326 93 4 , , PUNCT cana-1326 93 5 π‘‡π‘’βˆ—)β€– π‘‡π‘’βˆ—)β€– PROPN cana-1326 93 6 β€–1 β€–1 NOUN cana-1326 93 7 + + CCONJ cana-1326 93 8 𝑑(𝑒 𝑑(𝑒 ADJ cana-1326 93 9 , , PUNCT cana-1326 93 10 π‘’βˆ—)β€– π‘’βˆ—)β€– ADJ cana-1326 93 11 + + CCONJ cana-1326 93 12 𝐢√2 𝐢√2 PROPN cana-1326 93 13 ‖𝑑(π‘’βˆ— ‖𝑑(π‘’βˆ— NOUN cana-1326 93 14 , , PUNCT cana-1326 93 15 𝑆𝑒)‖‖𝑑(𝑒 𝑆𝑒)‖‖𝑑(𝑒 NOUN cana-1326 93 16 , , PUNCT cana-1326 93 17 π‘‡π‘’βˆ—)β€– π‘‡π‘’βˆ—)β€– PROPN cana-1326 93 18 β€–1 β€–1 NOUN cana-1326 93 19 + + CCONJ cana-1326 93 20 𝑑(𝑒 𝑑(𝑒 ADJ cana-1326 93 21 , , PUNCT cana-1326 93 22 π‘’βˆ—)β€– π‘’βˆ—)β€– PRON cana-1326 93 23 + + CCONJ cana-1326 93 24 𝐷√2 𝐷√2 PROPN cana-1326 93 25 ‖𝑑(𝑒 ‖𝑑(𝑒 NOUN cana-1326 93 26 , , PUNCT cana-1326 93 27 𝑆𝑒)‖‖𝑑(𝑒 𝑆𝑒)‖‖𝑑(𝑒 NOUN cana-1326 93 28 , , PUNCT cana-1326 93 29 π‘‡π‘’βˆ—)β€– π‘‡π‘’βˆ—)β€– PROPN cana-1326 93 30 β€–1 β€–1 NOUN cana-1326 93 31 + + CCONJ cana-1326 93 32 𝑑(𝑒 𝑑(𝑒 ADJ cana-1326 93 33 , , PUNCT cana-1326 93 34 π‘’βˆ—)β€– π‘’βˆ—)β€– ADJ cana-1326 93 35 + + PUNCT cana-1326 93 36 𝐸 𝐸 PROPN cana-1326 93 37 ‖𝑑(π‘’βˆ— ‖𝑑(π‘’βˆ— NOUN cana-1326 93 38 , , PUNCT cana-1326 93 39 𝑆𝑒)‖‖𝑑(π‘’βˆ— 𝑆𝑒)‖‖𝑑(π‘’βˆ— PROPN cana-1326 93 40 , , PUNCT cana-1326 93 41 π‘‡π‘’βˆ—)β€– π‘‡π‘’βˆ—)β€– PROPN cana-1326 93 42 β€–1 β€–1 NOUN cana-1326 93 43 + + CCONJ cana-1326 93 44 𝑑(𝑒 𝑑(𝑒 ADJ cana-1326 93 45 , , PUNCT cana-1326 93 46 π‘’βˆ—)β€– π‘’βˆ—)β€– ADJ cana-1326 93 47 ≀ ≀ ADJ cana-1326 93 48 𝐴‖𝑑(𝑒 𝐴‖𝑑(𝑒 NOUN cana-1326 93 49 , , PUNCT cana-1326 93 50 π‘’βˆ—)β€– π‘’βˆ—)β€– PUNCT cana-1326 93 51 + + CCONJ cana-1326 93 52 𝐢√2‖𝑑(π‘’βˆ— 𝐢√2‖𝑑(π‘’βˆ— NOUN cana-1326 93 53 , , PUNCT cana-1326 93 54 𝑒)β€– 𝑒)β€– NOUN cana-1326 94 1 [ [ X cana-1326 94 2 as as ADP cana-1326 94 3 β€–1 β€–1 NOUN cana-1326 94 4 + + CCONJ cana-1326 94 5 𝑑(𝑒 𝑑(𝑒 ADJ cana-1326 94 6 , , PUNCT cana-1326 94 7 π‘’βˆ—)β€– π‘’βˆ—)β€– X cana-1326 94 8 > > X cana-1326 94 9 ‖𝑑(𝑒 ‖𝑑(𝑒 X cana-1326 94 10 , , PUNCT cana-1326 94 11 π‘’βˆ—)β€– π‘’βˆ—)β€– PROPN cana-1326 94 12 ] ] PUNCT cana-1326 94 13 ∴ ∴ PROPN cana-1326 94 14 ‖𝑑(𝑒 ‖𝑑(𝑒 NOUN cana-1326 94 15 , , PUNCT cana-1326 94 16 π‘’βˆ—)β€– π‘’βˆ—)β€– X cana-1326 94 17 ≀ ≀ X cana-1326 94 18 ( ( PUNCT cana-1326 94 19 𝐴 𝐴 PROPN cana-1326 94 20 + + CCONJ cana-1326 94 21 𝐢√2)‖𝑑(𝑒 𝐢√2)‖𝑑(𝑒 PROPN cana-1326 94 22 , , PUNCT cana-1326 94 23 π‘’βˆ—)β€– π‘’βˆ—)β€– X cana-1326 94 24 communications communication NOUN cana-1326 94 25 on on ADP cana-1326 94 26 applied apply VERB cana-1326 94 27 nonlinear nonlinear ADJ cana-1326 94 28 analysis analysis NOUN cana-1326 94 29 issn issn NOUN cana-1326 94 30 : : PUNCT cana-1326 94 31 1074 1074 NUM cana-1326 94 32 - - PUNCT cana-1326 94 33 133x 133x NUM cana-1326 94 34 vol vol NOUN cana-1326 94 35 31 31 NUM cana-1326 94 36 no no NOUN cana-1326 94 37 . . PUNCT cana-1326 95 1 7s 7 NOUN cana-1326 95 2 ( ( PUNCT cana-1326 95 3 2024 2024 NUM cana-1326 95 4 ) ) PUNCT cana-1326 95 5 471 471 NUM cana-1326 95 6 https://internationalpubls.com https://internationalpubls.com X cana-1326 96 1 this this PRON cana-1326 96 2 is be AUX cana-1326 96 3 a a DET cana-1326 96 4 contradiction contradiction NOUN cana-1326 96 5 because because SCONJ cana-1326 96 6 ( ( PUNCT cana-1326 96 7 𝐴 𝐴 PROPN cana-1326 96 8 + + CCONJ cana-1326 96 9 𝐢√2 𝐢√2 PROPN cana-1326 96 10 ) ) PUNCT cana-1326 96 11 < < X cana-1326 96 12 1 1 X cana-1326 96 13 . . X cana-1326 96 14 hence hence ADV cana-1326 96 15 𝑒 𝑒 X cana-1326 96 16 = = NOUN cana-1326 96 17 π‘’βˆ—. π‘’βˆ—. X cana-1326 96 18 therefore therefore ADV cana-1326 96 19 u u NOUN cana-1326 96 20 is be AUX cana-1326 96 21 a a DET cana-1326 96 22 unique unique ADJ cana-1326 96 23 common common ADJ cana-1326 96 24 fixed fix VERB cana-1326 96 25 point point NOUN cana-1326 96 26 of of ADP cana-1326 96 27 t t PROPN cana-1326 96 28 and and CCONJ cana-1326 96 29 s. s. PROPN cana-1326 96 30 this this PRON cana-1326 96 31 completes complete VERB cana-1326 96 32 the the DET cana-1326 96 33 proof proof NOUN cana-1326 96 34 of of ADP cana-1326 96 35 the the DET cana-1326 96 36 theorem theorem NOUN cana-1326 96 37 . . PUNCT cana-1326 97 1 remark remark VERB cana-1326 97 2 1.1 1.1 NUM cana-1326 97 3 : : PUNCT cana-1326 97 4 the the DET cana-1326 97 5 result result NOUN cana-1326 97 6 of of ADP cana-1326 97 7 theorem theorem ADJ cana-1326 97 8 1.1 1.1 NUM cana-1326 97 9 remains remain VERB cana-1326 97 10 true true ADJ cana-1326 97 11 if if SCONJ cana-1326 97 12 the the DET cana-1326 97 13 condition condition NOUN cana-1326 97 14 ( ( PUNCT cana-1326 97 15 1.2 1.2 NUM cana-1326 97 16 ) ) PUNCT cana-1326 97 17 is be AUX cana-1326 97 18 replaced replace VERB cana-1326 97 19 by by ADP cana-1326 97 20 the the DET cana-1326 97 21 condition condition NOUN cana-1326 97 22 ‖𝑑(π‘₯0 ‖𝑑(π‘₯0 PROPN cana-1326 97 23 , , PUNCT cana-1326 97 24 𝑇π‘₯0β€– 𝑇π‘₯0β€– PUNCT cana-1326 97 25 ≀ ≀ X cana-1326 97 26 ( ( PUNCT cana-1326 97 27 1 1 NUM cana-1326 97 28 βˆ’ βˆ’ PROPN cana-1326 97 29 πœ†)|π‘Ÿ| πœ†)|π‘Ÿ| PROPN cana-1326 97 30 . . PUNCT cana-1326 98 1 corollary corollary ADJ cana-1326 98 2 1.1 1.1 NUM cana-1326 98 3 : : PUNCT cana-1326 98 4 let let VERB cana-1326 98 5 ( ( PUNCT cana-1326 98 6 x x NOUN cana-1326 98 7 , , PUNCT cana-1326 98 8 d d NOUN cana-1326 98 9 ) ) PUNCT cana-1326 98 10 be be AUX cana-1326 98 11 a a DET cana-1326 98 12 complete complete ADJ cana-1326 98 13 bi bi ADJ cana-1326 98 14 - - ADJ cana-1326 98 15 complex complex ADJ cana-1326 98 16 valued value VERB cana-1326 98 17 b b NOUN cana-1326 98 18 - - PUNCT cana-1326 98 19 metric metric ADJ cana-1326 98 20 space space NOUN cana-1326 98 21 with with ADP cana-1326 98 22 coefficient coefficient NOUN cana-1326 98 23 sβ‰₯ sβ‰₯ PROPN cana-1326 98 24 1 1 NUM cana-1326 98 25 and and CCONJ cana-1326 98 26 degenerated degenerated ADJ cana-1326 98 27 1+d(x 1+d(x NUM cana-1326 98 28 , , PUNCT cana-1326 98 29 y y PROPN cana-1326 98 30 ) ) PUNCT cana-1326 98 31 , , PUNCT cana-1326 98 32 β€–1 β€–1 NOUN cana-1326 99 1 + + CCONJ cana-1326 99 2 d(x d(x PROPN cana-1326 99 3 , , PUNCT cana-1326 99 4 y) y) PUNCT cana-1326 99 5 οΏ½ οΏ½ PROPN cana-1326 99 6 β€– β€– ADJ cana-1326 99 7 β‰  β‰  PROPN cana-1326 99 8 0 0 NUM cana-1326 99 9 οΏ½ οΏ½ NOUN cana-1326 99 10 and and CCONJ cana-1326 99 11 π‘₯0 π‘₯0 PROPN cana-1326 99 12 ∈ ∈ PROPN cana-1326 99 13 𝑋. 𝑋. PROPN cana-1326 99 14 let let VERB cana-1326 99 15 0 0 NUM cana-1326 99 16 ≀ ≀ NUM cana-1326 100 1 π‘Ÿ π‘Ÿ DET cana-1326 100 2 ∈ ∈ PROPN cana-1326 100 3 β„‚ β„‚ PROPN cana-1326 100 4 and and CCONJ cana-1326 100 5 a a DET cana-1326 100 6 , , PUNCT cana-1326 100 7 b b NOUN cana-1326 100 8 , , PUNCT cana-1326 100 9 c c NOUN cana-1326 100 10 , , PUNCT cana-1326 100 11 d d X cana-1326 100 12 are be AUX cana-1326 100 13 non non ADJ cana-1326 100 14 negative negative ADJ cana-1326 100 15 reals real NOUN cana-1326 100 16 such such ADJ cana-1326 100 17 that that SCONJ cana-1326 100 18 a+√2𝐡 a+√2𝐡 PROPN cana-1326 101 1 + + CCONJ cana-1326 101 2 √2𝐢 √2𝐢 PRON cana-1326 101 3 + + NOUN cana-1326 101 4 √2𝑠𝐷 √2𝑠𝐷 NOUN cana-1326 101 5 < < X cana-1326 101 6 1 1 X cana-1326 101 7 . . PUNCT cana-1326 102 1 let let VERB cana-1326 102 2 s s NOUN cana-1326 102 3 , , PUNCT cana-1326 102 4 t t PROPN cana-1326 102 5 : : PUNCT cana-1326 102 6 xβ†’x xβ†’x NUM cana-1326 102 7 are be AUX cana-1326 102 8 mappings mapping NOUN cana-1326 102 9 satisfying satisfying ADJ cana-1326 102 10 : : PUNCT cana-1326 103 1 d(sx d(sx NOUN cana-1326 103 2 , , PUNCT cana-1326 103 3 ty ty NUM cana-1326 103 4 ) ) PUNCT cana-1326 103 5 ≲𝑖2ad(x ≲𝑖2ad(x ADV cana-1326 103 6 , , PUNCT cana-1326 103 7 y)+b y)+b PROPN cana-1326 103 8 𝑑(π‘₯,𝑆π‘₯)𝑑(𝑦,𝑇𝑦 𝑑(π‘₯,𝑆π‘₯)𝑑(𝑦,𝑇𝑦 NUM cana-1326 103 9 ) ) PUNCT cana-1326 103 10 1+𝑑(π‘₯,𝑦 1+𝑑(π‘₯,𝑦 NUM cana-1326 103 11 ) ) PUNCT cana-1326 104 1 + + CCONJ cana-1326 104 2 𝐢 𝐢 PROPN cana-1326 104 3 𝑑(𝑦,𝑆π‘₯)𝑑(π‘₯,𝑇𝑦 𝑑(𝑦,𝑆π‘₯)𝑑(π‘₯,𝑇𝑦 NOUN cana-1326 104 4 ) ) PUNCT cana-1326 104 5 1+𝑑(π‘₯,𝑦 1+𝑑(π‘₯,𝑦 NOUN cana-1326 104 6 ) ) PUNCT cana-1326 105 1 + + NUM cana-1326 105 2 𝐷 𝐷 PROPN cana-1326 105 3 𝑑(π‘₯,𝑆π‘₯)𝑑(π‘₯,𝑇𝑦 𝑑(π‘₯,𝑆π‘₯)𝑑(π‘₯,𝑇𝑦 NOUN cana-1326 105 4 ) ) PUNCT cana-1326 105 5 1+𝑑(π‘₯,𝑦 1+𝑑(π‘₯,𝑦 X cana-1326 105 6 ) ) PUNCT cana-1326 105 7 οΏ½ οΏ½ PROPN cana-1326 105 8 οΏ½ οΏ½ PROPN cana-1326 105 9 οΏ½ οΏ½ PROPN cana-1326 105 10 οΏ½ οΏ½ PROPN cana-1326 105 11 οΏ½ οΏ½ PROPN cana-1326 105 12 οΏ½ οΏ½ PROPN cana-1326 105 13 οΏ½ οΏ½ PROPN cana-1326 105 14 οΏ½ οΏ½ PROPN cana-1326 105 15 οΏ½ οΏ½ PROPN cana-1326 105 16 οΏ½ οΏ½ PROPN cana-1326 105 17 οΏ½ οΏ½ PROPN cana-1326 105 18 οΏ½ οΏ½ PROPN cana-1326 105 19 οΏ½ οΏ½ PROPN cana-1326 105 20 οΏ½ οΏ½ PROPN cana-1326 105 21 οΏ½ οΏ½ PROPN cana-1326 105 22 οΏ½ οΏ½ PROPN cana-1326 105 23 οΏ½ οΏ½ PROPN cana-1326 105 24 οΏ½ οΏ½ PROPN cana-1326 105 25 οΏ½ οΏ½ PROPN cana-1326 105 26 οΏ½ οΏ½ PROPN cana-1326 105 27 οΏ½ οΏ½ PROPN cana-1326 105 28 οΏ½ οΏ½ PROPN cana-1326 105 29 οΏ½ οΏ½ PROPN cana-1326 105 30 οΏ½ οΏ½ PROPN cana-1326 105 31 οΏ½ οΏ½ PROPN cana-1326 105 32 οΏ½ οΏ½ PROPN cana-1326 105 33 οΏ½ οΏ½ PROPN cana-1326 105 34 οΏ½ οΏ½ PROPN cana-1326 105 35 οΏ½ οΏ½ PROPN cana-1326 105 36 οΏ½ οΏ½ PROPN cana-1326 105 37 οΏ½ οΏ½ PROPN cana-1326 105 38 οΏ½ οΏ½ PROPN cana-1326 105 39 οΏ½ οΏ½ PROPN cana-1326 105 40 οΏ½ οΏ½ PROPN cana-1326 105 41 οΏ½ οΏ½ PROPN cana-1326 105 42 οΏ½ οΏ½ PROPN cana-1326 105 43 οΏ½ οΏ½ PROPN cana-1326 105 44 οΏ½ οΏ½ PROPN cana-1326 105 45 οΏ½ οΏ½ PROPN cana-1326 105 46 οΏ½ οΏ½ PROPN cana-1326 105 47 for for ADP cana-1326 105 48 all all DET cana-1326 105 49 x x ADJ cana-1326 105 50 , , PUNCT cana-1326 105 51 y∈ y∈ PROPN cana-1326 105 52 b(π‘₯0 b(π‘₯0 NOUN cana-1326 105 53 , , PUNCT cana-1326 105 54 π‘Ÿ)Μ…Μ… π‘Ÿ)Μ…Μ… PROPN cana-1326 105 55 Μ…Μ… Μ…Μ… PROPN cana-1326 105 56 Μ…Μ… Μ…Μ… PROPN cana-1326 105 57 Μ…Μ… Μ…Μ… PROPN cana-1326 105 58 Μ…Μ… Μ…Μ… PROPN cana-1326 105 59 . . PUNCT cana-1326 106 1 if if SCONJ cana-1326 106 2 ‖𝑑(π‘₯0 ‖𝑑(π‘₯0 PROPN cana-1326 106 3 , , PUNCT cana-1326 106 4 𝑆π‘₯0β€– 𝑆π‘₯0β€– PROPN cana-1326 106 5 ≀ ≀ X cana-1326 106 6 ( ( PUNCT cana-1326 106 7 1 1 NUM cana-1326 106 8 βˆ’ βˆ’ PROPN cana-1326 106 9 πœ†)|π‘Ÿ| πœ†)|π‘Ÿ| PROPN cana-1326 106 10 , , PUNCT cana-1326 106 11 where where SCONJ cana-1326 106 12 Ξ»= Ξ»= ADJ cana-1326 106 13 max max PROPN cana-1326 106 14 { { PUNCT cana-1326 106 15 𝐴+√2𝑠𝐷 𝐴+√2𝑠𝐷 PROPN cana-1326 106 16 1βˆ’βˆš2π΅βˆ’βˆš2𝑠𝐷 1βˆ’βˆš2π΅βˆ’βˆš2𝑠𝐷 NOUN cana-1326 106 17 , , PUNCT cana-1326 106 18 𝐴 𝐴 PROPN cana-1326 106 19 1βˆ’βˆš2𝐡 1βˆ’βˆš2𝐡 PROPN cana-1326 107 1 + + PROPN cana-1326 107 2 , , PUNCT cana-1326 107 3 then then ADV cana-1326 107 4 there there PRON cana-1326 107 5 exist exist VERB cana-1326 107 6 a a DET cana-1326 107 7 unique unique ADJ cana-1326 107 8 point point NOUN cana-1326 107 9 u∈ u∈ NOUN cana-1326 107 10 b(π‘₯0 b(π‘₯0 NOUN cana-1326 107 11 , , PUNCT cana-1326 107 12 π‘Ÿ)Μ…Μ… π‘Ÿ)Μ…Μ… PROPN cana-1326 107 13 Μ…Μ… Μ…Μ… PROPN cana-1326 107 14 Μ…Μ… Μ…Μ… PROPN cana-1326 107 15 Μ…Μ… Μ…Μ… PROPN cana-1326 107 16 Μ…Μ… Μ…Μ… PROPN cana-1326 108 1 such such ADJ cana-1326 108 2 that that SCONJ cana-1326 108 3 u u PROPN cana-1326 108 4 = = PROPN cana-1326 108 5 su su PROPN cana-1326 108 6 = = PROPN cana-1326 108 7 tu tu PROPN cana-1326 108 8 . . PUNCT cana-1326 108 9 proof proof NOUN cana-1326 108 10 : : PUNCT cana-1326 108 11 we we PRON cana-1326 108 12 can can AUX cana-1326 108 13 prove prove VERB cana-1326 108 14 this this DET cana-1326 108 15 result result NOUN cana-1326 108 16 by by ADP cana-1326 108 17 applying apply VERB cana-1326 108 18 theorem theorem ADJ cana-1326 108 19 1.1 1.1 NUM cana-1326 108 20 by by ADP cana-1326 108 21 setting set VERB cana-1326 108 22 e=0 e=0 PROPN cana-1326 108 23 . . PUNCT cana-1326 109 1 corollary corollary ADJ cana-1326 109 2 1.2 1.2 NUM cana-1326 110 1 : : PUNCT cana-1326 110 2 let let AUX cana-1326 110 3 ( ( PUNCT cana-1326 110 4 x x X cana-1326 110 5 , , PUNCT cana-1326 110 6 d d NOUN cana-1326 110 7 ) ) PUNCT cana-1326 110 8 be be AUX cana-1326 110 9 a a DET cana-1326 110 10 complete complete ADJ cana-1326 110 11 bi bi ADJ cana-1326 110 12 - - ADJ cana-1326 110 13 complex complex ADJ cana-1326 110 14 valued value VERB cana-1326 110 15 b b NOUN cana-1326 110 16 - - PUNCT cana-1326 110 17 metric metric ADJ cana-1326 110 18 space space NOUN cana-1326 110 19 with with ADP cana-1326 110 20 coefficient coefficient NOUN cana-1326 110 21 sβ‰₯ sβ‰₯ PROPN cana-1326 110 22 1 1 NUM cana-1326 110 23 and and CCONJ cana-1326 110 24 degenerated degenerated ADJ cana-1326 110 25 1+d(x 1+d(x NUM cana-1326 110 26 , , PUNCT cana-1326 110 27 y y PROPN cana-1326 110 28 ) ) PUNCT cana-1326 110 29 , , PUNCT cana-1326 110 30 β€–1 β€–1 NOUN cana-1326 111 1 + + CCONJ cana-1326 111 2 d(x d(x PROPN cana-1326 111 3 , , PUNCT cana-1326 111 4 y) y) PUNCT cana-1326 111 5 οΏ½ οΏ½ PROPN cana-1326 111 6 β€– β€– ADJ cana-1326 111 7 β‰  β‰  PROPN cana-1326 111 8 0 0 NUM cana-1326 111 9 οΏ½ οΏ½ NOUN cana-1326 111 10 and and CCONJ cana-1326 111 11 π‘₯0 π‘₯0 PROPN cana-1326 111 12 ∈ ∈ PROPN cana-1326 111 13 𝑋. 𝑋. PROPN cana-1326 111 14 let let VERB cana-1326 111 15 0 0 NUM cana-1326 111 16 ≀ ≀ NUM cana-1326 112 1 π‘Ÿ π‘Ÿ DET cana-1326 112 2 ∈ ∈ PROPN cana-1326 112 3 β„‚ β„‚ PROPN cana-1326 112 4 and and CCONJ cana-1326 112 5 a a DET cana-1326 112 6 , , PUNCT cana-1326 112 7 b b NOUN cana-1326 112 8 , , PUNCT cana-1326 112 9 c c PROPN cana-1326 112 10 and and CCONJ cana-1326 112 11 e e PROPN cana-1326 112 12 are be AUX cana-1326 112 13 non non ADJ cana-1326 112 14 negative negative ADJ cana-1326 112 15 reals real NOUN cana-1326 112 16 such such ADJ cana-1326 112 17 that that SCONJ cana-1326 112 18 a+√2𝐡 a+√2𝐡 PROPN cana-1326 113 1 + + CCONJ cana-1326 113 2 √2𝐢 √2𝐢 PRON cana-1326 113 3 + + CCONJ cana-1326 113 4 √2𝑠𝐸 √2𝑠𝐸 NOUN cana-1326 113 5 < < X cana-1326 113 6 1 1 X cana-1326 113 7 . . PUNCT cana-1326 114 1 let let VERB cana-1326 114 2 s s NOUN cana-1326 114 3 , , PUNCT cana-1326 114 4 t t PROPN cana-1326 114 5 : : PUNCT cana-1326 114 6 xβ†’x xβ†’x NUM cana-1326 114 7 are be AUX cana-1326 114 8 mappings mapping NOUN cana-1326 114 9 satisfying satisfying ADJ cana-1326 114 10 : : PUNCT cana-1326 115 1 d(sx d(sx NOUN cana-1326 115 2 , , PUNCT cana-1326 115 3 ty ty NUM cana-1326 115 4 ) ) PUNCT cana-1326 115 5 ≲𝑖2ad(x ≲𝑖2ad(x ADV cana-1326 115 6 , , PUNCT cana-1326 115 7 y)+b y)+b PROPN cana-1326 115 8 𝑑(π‘₯,𝑆π‘₯)𝑑(𝑦,𝑇𝑦 𝑑(π‘₯,𝑆π‘₯)𝑑(𝑦,𝑇𝑦 NUM cana-1326 115 9 ) ) PUNCT cana-1326 115 10 1+𝑑(π‘₯,𝑦 1+𝑑(π‘₯,𝑦 NUM cana-1326 115 11 ) ) PUNCT cana-1326 116 1 + + CCONJ cana-1326 116 2 𝐢 𝐢 PROPN cana-1326 116 3 𝑑(𝑦,𝑆π‘₯)𝑑(π‘₯,𝑇𝑦 𝑑(𝑦,𝑆π‘₯)𝑑(π‘₯,𝑇𝑦 NOUN cana-1326 116 4 ) ) PUNCT cana-1326 116 5 1+𝑑(π‘₯,𝑦 1+𝑑(π‘₯,𝑦 NOUN cana-1326 116 6 ) ) PUNCT cana-1326 117 1 + + CCONJ cana-1326 117 2 𝐸 𝐸 PROPN cana-1326 117 3 𝑑(𝑦,𝑆π‘₯)𝑑(𝑦,𝑇𝑦 𝑑(𝑦,𝑆π‘₯)𝑑(𝑦,𝑇𝑦 NOUN cana-1326 117 4 ) ) PUNCT cana-1326 117 5 1+𝑑(π‘₯,𝑦 1+𝑑(π‘₯,𝑦 NOUN cana-1326 117 6 ) ) PUNCT cana-1326 117 7 οΏ½ οΏ½ PROPN cana-1326 117 8 οΏ½ οΏ½ PROPN cana-1326 117 9 οΏ½ οΏ½ PROPN cana-1326 117 10 οΏ½ οΏ½ PROPN cana-1326 117 11 οΏ½ οΏ½ PROPN cana-1326 117 12 οΏ½ οΏ½ PROPN cana-1326 117 13 οΏ½ οΏ½ PROPN cana-1326 117 14 οΏ½ οΏ½ PROPN cana-1326 117 15 οΏ½ οΏ½ PROPN cana-1326 117 16 οΏ½ οΏ½ PROPN cana-1326 117 17 οΏ½ οΏ½ PROPN cana-1326 117 18 οΏ½ οΏ½ PROPN cana-1326 117 19 οΏ½ οΏ½ PROPN cana-1326 117 20 οΏ½ οΏ½ PROPN cana-1326 117 21 οΏ½ οΏ½ PROPN cana-1326 117 22 οΏ½ οΏ½ PROPN cana-1326 117 23 οΏ½ οΏ½ PROPN cana-1326 117 24 οΏ½ οΏ½ PROPN cana-1326 117 25 οΏ½ οΏ½ PROPN cana-1326 117 26 οΏ½ οΏ½ PROPN cana-1326 117 27 οΏ½ οΏ½ PROPN cana-1326 117 28 οΏ½ οΏ½ PROPN cana-1326 117 29 οΏ½ οΏ½ PROPN cana-1326 117 30 οΏ½ οΏ½ PROPN cana-1326 117 31 οΏ½ οΏ½ PROPN cana-1326 117 32 οΏ½ οΏ½ PROPN cana-1326 117 33 οΏ½ οΏ½ PROPN cana-1326 117 34 οΏ½ οΏ½ PROPN cana-1326 117 35 οΏ½ οΏ½ PROPN cana-1326 117 36 οΏ½ οΏ½ PROPN cana-1326 117 37 οΏ½ οΏ½ PROPN cana-1326 117 38 οΏ½ οΏ½ PROPN cana-1326 117 39 οΏ½ οΏ½ PROPN cana-1326 117 40 οΏ½ οΏ½ PROPN cana-1326 117 41 οΏ½ οΏ½ PROPN cana-1326 117 42 οΏ½ οΏ½ PROPN cana-1326 117 43 οΏ½ οΏ½ PROPN cana-1326 117 44 οΏ½ οΏ½ PROPN cana-1326 117 45 οΏ½ οΏ½ PROPN cana-1326 117 46 οΏ½ οΏ½ PROPN cana-1326 117 47 for for ADP cana-1326 117 48 all all DET cana-1326 117 49 x x ADJ cana-1326 117 50 , , PUNCT cana-1326 117 51 y∈ y∈ PROPN cana-1326 117 52 b(π‘₯0 b(π‘₯0 NOUN cana-1326 117 53 , , PUNCT cana-1326 117 54 π‘Ÿ)Μ…Μ… π‘Ÿ)Μ…Μ… PROPN cana-1326 117 55 Μ…Μ… Μ…Μ… PROPN cana-1326 117 56 Μ…Μ… Μ…Μ… PROPN cana-1326 117 57 Μ…Μ… Μ…Μ… PROPN cana-1326 117 58 Μ…Μ… Μ…Μ… PROPN cana-1326 117 59 . . PUNCT cana-1326 118 1 if if SCONJ cana-1326 118 2 ‖𝑑(π‘₯0 ‖𝑑(π‘₯0 PROPN cana-1326 118 3 , , PUNCT cana-1326 118 4 𝑆π‘₯0β€– 𝑆π‘₯0β€– PROPN cana-1326 118 5 ≀ ≀ X cana-1326 118 6 ( ( PUNCT cana-1326 118 7 1 1 NUM cana-1326 118 8 βˆ’ βˆ’ PROPN cana-1326 118 9 πœ†)|π‘Ÿ| πœ†)|π‘Ÿ| PROPN cana-1326 118 10 , , PUNCT cana-1326 118 11 where where SCONJ cana-1326 118 12 Ξ»= Ξ»= ADJ cana-1326 118 13 max max PROPN cana-1326 118 14 { { PUNCT cana-1326 118 15 𝐴 𝐴 PROPN cana-1326 118 16 1βˆ’βˆš2𝐡 1βˆ’βˆš2𝐡 PROPN cana-1326 118 17 , , PUNCT cana-1326 118 18 𝐴+√2𝑠𝐸 𝐴+√2𝑠𝐸 PROPN cana-1326 118 19 1βˆ’βˆš2π΅βˆ’βˆš2𝑠𝐸 1βˆ’βˆš2π΅βˆ’βˆš2𝑠𝐸 NUM cana-1326 118 20 + + ADJ cana-1326 118 21 , , PUNCT cana-1326 118 22 then then ADV cana-1326 118 23 there there PRON cana-1326 118 24 exist exist VERB cana-1326 118 25 a a DET cana-1326 118 26 unique unique ADJ cana-1326 118 27 point point NOUN cana-1326 118 28 u∈ u∈ NOUN cana-1326 118 29 b(π‘₯0 b(π‘₯0 NOUN cana-1326 118 30 , , PUNCT cana-1326 118 31 π‘Ÿ)Μ…Μ… π‘Ÿ)Μ…Μ… PROPN cana-1326 118 32 Μ…Μ… Μ…Μ… PROPN cana-1326 118 33 Μ…Μ… Μ…Μ… PROPN cana-1326 118 34 Μ…Μ… Μ…Μ… PROPN cana-1326 118 35 Μ…Μ… Μ…Μ… PROPN cana-1326 119 1 such such ADJ cana-1326 119 2 that that SCONJ cana-1326 119 3 u u PROPN cana-1326 119 4 = = PROPN cana-1326 119 5 su su PROPN cana-1326 119 6 = = PROPN cana-1326 119 7 tu tu PROPN cana-1326 119 8 . . PUNCT cana-1326 119 9 proof proof NOUN cana-1326 119 10 : : PUNCT cana-1326 119 11 we we PRON cana-1326 119 12 can can AUX cana-1326 119 13 prove prove VERB cana-1326 119 14 this this DET cana-1326 119 15 result result NOUN cana-1326 119 16 by by ADP cana-1326 119 17 applying apply VERB cana-1326 119 18 theorem theorem NOUN cana-1326 119 19 1.1 1.1 NUM cana-1326 119 20 by by ADP cana-1326 119 21 setting set VERB cana-1326 119 22 d=0 d=0 PROPN cana-1326 119 23 . . PUNCT cana-1326 120 1 corollary corollary NOUN cana-1326 120 2 1.3 1.3 NUM cana-1326 120 3 : : PUNCT cana-1326 120 4 let let AUX cana-1326 120 5 ( ( PUNCT cana-1326 120 6 x x X cana-1326 120 7 , , PUNCT cana-1326 120 8 d d NOUN cana-1326 120 9 ) ) PUNCT cana-1326 120 10 be be AUX cana-1326 120 11 a a DET cana-1326 120 12 complete complete ADJ cana-1326 120 13 bi bi ADJ cana-1326 120 14 - - ADJ cana-1326 120 15 complex complex ADJ cana-1326 120 16 valued value VERB cana-1326 120 17 b b NOUN cana-1326 120 18 - - PUNCT cana-1326 120 19 metric metric ADJ cana-1326 120 20 space space NOUN cana-1326 120 21 with with ADP cana-1326 120 22 coefficient coefficient NOUN cana-1326 120 23 sβ‰₯ sβ‰₯ PROPN cana-1326 120 24 1 1 NUM cana-1326 120 25 and and CCONJ cana-1326 120 26 degenerated degenerated ADJ cana-1326 120 27 1+d(x 1+d(x NUM cana-1326 120 28 , , PUNCT cana-1326 120 29 y y PROPN cana-1326 120 30 ) ) PUNCT cana-1326 120 31 , , PUNCT cana-1326 120 32 β€–1 β€–1 NOUN cana-1326 121 1 + + CCONJ cana-1326 121 2 d(x d(x PROPN cana-1326 121 3 , , PUNCT cana-1326 121 4 y) y) PUNCT cana-1326 121 5 οΏ½ οΏ½ PROPN cana-1326 121 6 β€– β€– ADJ cana-1326 121 7 β‰  β‰  PROPN cana-1326 121 8 0 0 NUM cana-1326 121 9 οΏ½ οΏ½ NOUN cana-1326 121 10 and and CCONJ cana-1326 121 11 π‘₯0 π‘₯0 PROPN cana-1326 121 12 ∈ ∈ PROPN cana-1326 121 13 𝑋. 𝑋. PROPN cana-1326 121 14 let let VERB cana-1326 121 15 0 0 NUM cana-1326 121 16 ≀ ≀ NUM cana-1326 122 1 π‘Ÿ π‘Ÿ DET cana-1326 122 2 ∈ ∈ PROPN cana-1326 122 3 β„‚ β„‚ PROPN cana-1326 122 4 and and CCONJ cana-1326 122 5 a a DET cana-1326 122 6 , , PUNCT cana-1326 122 7 b b NOUN cana-1326 122 8 , , PUNCT cana-1326 122 9 c c X cana-1326 122 10 be be AUX cana-1326 122 11 three three NUM cana-1326 122 12 non non ADJ cana-1326 122 13 negative negative ADJ cana-1326 122 14 reals real NOUN cana-1326 122 15 such such ADJ cana-1326 122 16 that that SCONJ cana-1326 122 17 a+√2𝐡 a+√2𝐡 PROPN cana-1326 123 1 + + CCONJ cana-1326 123 2 √2𝐢 √2𝐢 NOUN cana-1326 123 3 < < X cana-1326 123 4 1 1 NUM cana-1326 123 5 . . PUNCT cana-1326 124 1 let let VERB cana-1326 124 2 s s NOUN cana-1326 124 3 , , PUNCT cana-1326 124 4 t t PROPN cana-1326 124 5 : : PUNCT cana-1326 124 6 xβ†’x xβ†’x NUM cana-1326 124 7 are be AUX cana-1326 124 8 mappings mapping NOUN cana-1326 124 9 satisfying satisfying ADJ cana-1326 124 10 : : PUNCT cana-1326 125 1 d(sx d(sx NOUN cana-1326 125 2 , , PUNCT cana-1326 125 3 ty ty NUM cana-1326 125 4 ) ) PUNCT cana-1326 125 5 ≲𝑖2ad(x ≲𝑖2ad(x ADV cana-1326 125 6 , , PUNCT cana-1326 125 7 y)+b y)+b PROPN cana-1326 125 8 𝑑(π‘₯,𝑆π‘₯)𝑑(𝑦,𝑇𝑦 𝑑(π‘₯,𝑆π‘₯)𝑑(𝑦,𝑇𝑦 NUM cana-1326 125 9 ) ) PUNCT cana-1326 125 10 1+𝑑(π‘₯,𝑦 1+𝑑(π‘₯,𝑦 NUM cana-1326 125 11 ) ) PUNCT cana-1326 126 1 + + CCONJ cana-1326 126 2 𝐢 𝐢 PROPN cana-1326 126 3 𝑑(𝑦,𝑆π‘₯)𝑑(π‘₯,𝑇𝑦 𝑑(𝑦,𝑆π‘₯)𝑑(π‘₯,𝑇𝑦 NOUN cana-1326 126 4 ) ) PUNCT cana-1326 126 5 1+𝑑(π‘₯,𝑦 1+𝑑(π‘₯,𝑦 NOUN cana-1326 126 6 ) ) PUNCT cana-1326 126 7 οΏ½ οΏ½ PROPN cana-1326 126 8 οΏ½ οΏ½ PROPN cana-1326 126 9 οΏ½ οΏ½ PROPN cana-1326 126 10 οΏ½ οΏ½ PROPN cana-1326 126 11 οΏ½ οΏ½ PROPN cana-1326 126 12 οΏ½ οΏ½ PROPN cana-1326 126 13 οΏ½ οΏ½ PROPN cana-1326 126 14 οΏ½ οΏ½ PROPN cana-1326 126 15 οΏ½ οΏ½ PROPN cana-1326 126 16 οΏ½ οΏ½ PROPN cana-1326 126 17 οΏ½ οΏ½ PROPN cana-1326 126 18 οΏ½ οΏ½ PROPN cana-1326 126 19 οΏ½ οΏ½ PROPN cana-1326 126 20 οΏ½ οΏ½ PROPN cana-1326 126 21 οΏ½ οΏ½ PROPN cana-1326 126 22 οΏ½ οΏ½ PROPN cana-1326 126 23 οΏ½ οΏ½ PROPN cana-1326 126 24 οΏ½ οΏ½ PROPN cana-1326 126 25 οΏ½ οΏ½ PROPN cana-1326 126 26 οΏ½ οΏ½ PROPN cana-1326 126 27 οΏ½ οΏ½ PROPN cana-1326 126 28 οΏ½ οΏ½ PROPN cana-1326 126 29 οΏ½ οΏ½ PROPN cana-1326 126 30 οΏ½ οΏ½ PROPN cana-1326 126 31 οΏ½ οΏ½ PROPN cana-1326 126 32 οΏ½ οΏ½ PROPN cana-1326 126 33 οΏ½ οΏ½ PROPN cana-1326 126 34 οΏ½ οΏ½ PROPN cana-1326 126 35 οΏ½ οΏ½ PROPN cana-1326 126 36 οΏ½ οΏ½ PROPN cana-1326 126 37 οΏ½ οΏ½ PROPN cana-1326 126 38 οΏ½ οΏ½ PROPN cana-1326 126 39 οΏ½ οΏ½ PROPN cana-1326 126 40 οΏ½ οΏ½ PROPN cana-1326 126 41 οΏ½ οΏ½ PROPN cana-1326 126 42 οΏ½ οΏ½ PROPN cana-1326 126 43 οΏ½ οΏ½ PROPN cana-1326 126 44 οΏ½ οΏ½ PROPN cana-1326 126 45 οΏ½ οΏ½ PROPN cana-1326 126 46 for for ADP cana-1326 126 47 all all DET cana-1326 126 48 x x ADJ cana-1326 126 49 , , PUNCT cana-1326 126 50 y∈ y∈ PROPN cana-1326 126 51 b(π‘₯0 b(π‘₯0 NOUN cana-1326 126 52 , , PUNCT cana-1326 126 53 π‘Ÿ)Μ…Μ… π‘Ÿ)Μ…Μ… PROPN cana-1326 126 54 Μ…Μ… Μ…Μ… PROPN cana-1326 126 55 Μ…Μ… Μ…Μ… PROPN cana-1326 126 56 Μ…Μ… Μ…Μ… PROPN cana-1326 126 57 Μ…Μ… Μ…Μ… PROPN cana-1326 126 58 . . PUNCT cana-1326 127 1 if if SCONJ cana-1326 127 2 ‖𝑑(π‘₯0 ‖𝑑(π‘₯0 PROPN cana-1326 127 3 , , PUNCT cana-1326 127 4 𝑆π‘₯0β€– 𝑆π‘₯0β€– PROPN cana-1326 127 5 ≀ ≀ X cana-1326 127 6 ( ( PUNCT cana-1326 127 7 1 1 NUM cana-1326 127 8 βˆ’ βˆ’ PROPN cana-1326 127 9 πœ†)|π‘Ÿ| πœ†)|π‘Ÿ| PROPN cana-1326 127 10 , , PUNCT cana-1326 127 11 where where SCONJ cana-1326 127 12 Ξ»= Ξ»= VERB cana-1326 127 13 𝐴 𝐴 PROPN cana-1326 127 14 1βˆ’βˆš2𝐡 1βˆ’βˆš2𝐡 NOUN cana-1326 127 15 then then ADV cana-1326 127 16 there there PRON cana-1326 127 17 exist exist VERB cana-1326 127 18 a a DET cana-1326 127 19 unique unique ADJ cana-1326 127 20 point point NOUN cana-1326 127 21 u∈ u∈ NOUN cana-1326 127 22 b(π‘₯0 b(π‘₯0 NOUN cana-1326 127 23 , , PUNCT cana-1326 127 24 π‘Ÿ)Μ…Μ… π‘Ÿ)Μ…Μ… PROPN cana-1326 127 25 Μ…Μ… Μ…Μ… PROPN cana-1326 127 26 Μ…Μ… Μ…Μ… PROPN cana-1326 127 27 Μ…Μ… Μ…Μ… PROPN cana-1326 127 28 Μ…Μ… Μ…Μ… PROPN cana-1326 128 1 such such ADJ cana-1326 128 2 that that SCONJ cana-1326 128 3 u u PROPN cana-1326 128 4 = = PROPN cana-1326 128 5 su su PROPN cana-1326 128 6 = = PROPN cana-1326 128 7 tu tu PROPN cana-1326 128 8 . . PUNCT cana-1326 128 9 proof proof NOUN cana-1326 128 10 : : PUNCT cana-1326 128 11 we we PRON cana-1326 128 12 can can AUX cana-1326 128 13 prove prove VERB cana-1326 128 14 this this DET cana-1326 128 15 result result NOUN cana-1326 128 16 by by ADP cana-1326 128 17 applying apply VERB cana-1326 128 18 corollary corollary ADJ cana-1326 128 19 1.2 1.2 NUM cana-1326 128 20 by by ADP cana-1326 128 21 setting set VERB cana-1326 128 22 e=0 e=0 PROPN cana-1326 128 23 . . PUNCT cana-1326 129 1 corollary corollary ADJ cana-1326 129 2 1.4 1.4 NUM cana-1326 130 1 : : PUNCT cana-1326 130 2 let let AUX cana-1326 130 3 ( ( PUNCT cana-1326 130 4 x x X cana-1326 130 5 , , PUNCT cana-1326 130 6 d d NOUN cana-1326 130 7 ) ) PUNCT cana-1326 130 8 be be AUX cana-1326 130 9 a a DET cana-1326 130 10 complete complete ADJ cana-1326 130 11 bi bi ADJ cana-1326 130 12 - - ADJ cana-1326 130 13 complex complex ADJ cana-1326 130 14 valued value VERB cana-1326 130 15 b b NOUN cana-1326 130 16 - - PUNCT cana-1326 130 17 metric metric ADJ cana-1326 130 18 space space NOUN cana-1326 130 19 with with ADP cana-1326 130 20 coefficient coefficient NOUN cana-1326 130 21 sβ‰₯ sβ‰₯ PROPN cana-1326 130 22 1 1 NUM cana-1326 130 23 and and CCONJ cana-1326 130 24 degenerated degenerated ADJ cana-1326 130 25 1+d(x 1+d(x NUM cana-1326 130 26 , , PUNCT cana-1326 130 27 y y PROPN cana-1326 130 28 ) ) PUNCT cana-1326 130 29 , , PUNCT cana-1326 130 30 β€–1 β€–1 NOUN cana-1326 131 1 + + CCONJ cana-1326 131 2 d(x d(x PROPN cana-1326 131 3 , , PUNCT cana-1326 131 4 y) y) PUNCT cana-1326 131 5 οΏ½ οΏ½ PROPN cana-1326 131 6 β€– β€– ADJ cana-1326 131 7 β‰  β‰  PROPN cana-1326 131 8 0 0 NUM cana-1326 131 9 οΏ½ οΏ½ NOUN cana-1326 131 10 and and CCONJ cana-1326 131 11 π‘₯0 π‘₯0 PROPN cana-1326 131 12 ∈ ∈ PROPN cana-1326 131 13 𝑋. 𝑋. PROPN cana-1326 131 14 let let VERB cana-1326 131 15 0 0 NUM cana-1326 131 16 ≀ ≀ NUM cana-1326 132 1 π‘Ÿ π‘Ÿ DET cana-1326 132 2 ∈ ∈ PROPN cana-1326 132 3 β„‚ β„‚ PROPN cana-1326 132 4 and and CCONJ cana-1326 132 5 a a DET cana-1326 132 6 , , PUNCT cana-1326 132 7 b b NOUN cana-1326 132 8 are be AUX cana-1326 132 9 non non X cana-1326 132 10 negative negative ADJ cana-1326 132 11 reals real NOUN cana-1326 132 12 such such ADJ cana-1326 132 13 that that SCONJ cana-1326 132 14 a+√2𝐡 a+√2𝐡 PROPN cana-1326 132 15 < < X cana-1326 132 16 1 1 X cana-1326 132 17 . . PUNCT cana-1326 133 1 let let VERB cana-1326 133 2 s s NOUN cana-1326 133 3 , , PUNCT cana-1326 133 4 t t PROPN cana-1326 133 5 : : PUNCT cana-1326 133 6 xβ†’x xβ†’x NUM cana-1326 133 7 are be AUX cana-1326 133 8 mappings mapping NOUN cana-1326 133 9 satisfying satisfying ADJ cana-1326 133 10 : : PUNCT cana-1326 134 1 d(sx d(sx NOUN cana-1326 134 2 , , PUNCT cana-1326 134 3 ty ty NUM cana-1326 134 4 ) ) PUNCT cana-1326 134 5 ≲𝑖2ad(x ≲𝑖2ad(x ADV cana-1326 134 6 , , PUNCT cana-1326 134 7 y)+b y)+b PROPN cana-1326 134 8 𝑑(π‘₯,𝑆π‘₯)𝑑(𝑦,𝑇𝑦 𝑑(π‘₯,𝑆π‘₯)𝑑(𝑦,𝑇𝑦 NOUN cana-1326 134 9 ) ) PUNCT cana-1326 134 10 1+𝑑(π‘₯,𝑦 1+𝑑(π‘₯,𝑦 NOUN cana-1326 134 11 ) ) PUNCT cana-1326 134 12 οΏ½ οΏ½ PROPN cana-1326 134 13 οΏ½ οΏ½ PROPN cana-1326 134 14 οΏ½ οΏ½ PROPN cana-1326 134 15 οΏ½ οΏ½ PROPN cana-1326 134 16 οΏ½ οΏ½ PROPN cana-1326 134 17 for for ADP cana-1326 134 18 all all DET cana-1326 134 19 x x ADJ cana-1326 134 20 , , PUNCT cana-1326 134 21 y∈ y∈ PROPN cana-1326 134 22 b(π‘₯0 b(π‘₯0 NOUN cana-1326 134 23 , , PUNCT cana-1326 134 24 π‘Ÿ)Μ…Μ… π‘Ÿ)Μ…Μ… PROPN cana-1326 134 25 Μ…Μ… Μ…Μ… PROPN cana-1326 134 26 Μ…Μ… Μ…Μ… PROPN cana-1326 134 27 Μ…Μ… Μ…Μ… PROPN cana-1326 134 28 Μ…Μ… Μ…Μ… PROPN cana-1326 134 29 . . PUNCT cana-1326 135 1 if if SCONJ cana-1326 135 2 ‖𝑑(π‘₯0 ‖𝑑(π‘₯0 PROPN cana-1326 135 3 , , PUNCT cana-1326 135 4 𝑆π‘₯0β€– 𝑆π‘₯0β€– PROPN cana-1326 135 5 ≀ ≀ X cana-1326 135 6 ( ( PUNCT cana-1326 135 7 1 1 NUM cana-1326 135 8 βˆ’ βˆ’ PROPN cana-1326 135 9 πœ†)|π‘Ÿ| πœ†)|π‘Ÿ| PROPN cana-1326 135 10 , , PUNCT cana-1326 135 11 communications communication NOUN cana-1326 135 12 on on ADP cana-1326 135 13 applied apply VERB cana-1326 135 14 nonlinear nonlinear ADJ cana-1326 135 15 analysis analysis NOUN cana-1326 135 16 issn issn NOUN cana-1326 135 17 : : PUNCT cana-1326 135 18 1074 1074 NUM cana-1326 135 19 - - PUNCT cana-1326 135 20 133x 133x NUM cana-1326 135 21 vol vol NOUN cana-1326 135 22 31 31 NUM cana-1326 135 23 no no NOUN cana-1326 135 24 . . PUNCT cana-1326 136 1 7s 7 NOUN cana-1326 136 2 ( ( PUNCT cana-1326 136 3 2024 2024 NUM cana-1326 136 4 ) ) PUNCT cana-1326 136 5 472 472 NUM cana-1326 136 6 https://internationalpubls.com https://internationalpubls.com X cana-1326 136 7 where where SCONJ cana-1326 136 8 Ξ»= Ξ»= VERB cana-1326 136 9 𝐴 𝐴 PROPN cana-1326 136 10 1βˆ’βˆš2𝐡 1βˆ’βˆš2𝐡 NOUN cana-1326 136 11 then then ADV cana-1326 136 12 there there PRON cana-1326 136 13 exist exist VERB cana-1326 136 14 a a DET cana-1326 136 15 unique unique ADJ cana-1326 136 16 point point NOUN cana-1326 136 17 u∈ u∈ NOUN cana-1326 136 18 b(π‘₯0 b(π‘₯0 NOUN cana-1326 136 19 , , PUNCT cana-1326 136 20 π‘Ÿ)Μ…Μ… π‘Ÿ)Μ…Μ… PROPN cana-1326 136 21 Μ…Μ… Μ…Μ… PROPN cana-1326 136 22 Μ…Μ… Μ…Μ… PROPN cana-1326 136 23 Μ…Μ… Μ…Μ… PROPN cana-1326 136 24 Μ…Μ… Μ…Μ… PROPN cana-1326 137 1 such such ADJ cana-1326 137 2 that that SCONJ cana-1326 137 3 u u PROPN cana-1326 137 4 = = PROPN cana-1326 137 5 su su PROPN cana-1326 137 6 = = PROPN cana-1326 137 7 tu tu PROPN cana-1326 137 8 . . PUNCT cana-1326 137 9 proof proof NOUN cana-1326 137 10 : : PUNCT cana-1326 137 11 we we PRON cana-1326 137 12 can can AUX cana-1326 137 13 prove prove VERB cana-1326 137 14 this this DET cana-1326 137 15 result result NOUN cana-1326 137 16 by by ADP cana-1326 137 17 applying apply VERB cana-1326 137 18 corollary corollary ADJ cana-1326 137 19 1.3 1.3 NUM cana-1326 137 20 by by ADP cana-1326 137 21 setting set VERB cana-1326 137 22 c=0.our c=0.our PRON cana-1326 137 23 result result NOUN cana-1326 137 24 is be AUX cana-1326 137 25 the the DET cana-1326 137 26 extension extension NOUN cana-1326 137 27 of of ADP cana-1326 137 28 theorem theorem NOUN cana-1326 137 29 ( ( PUNCT cana-1326 137 30 3.1 3.1 NUM cana-1326 137 31 ) ) PUNCT cana-1326 137 32 of of ADP cana-1326 137 33 [ [ X cana-1326 137 34 6 6 NUM cana-1326 137 35 ] ] PUNCT cana-1326 137 36 to to ADP cana-1326 137 37 the the DET cana-1326 137 38 closed closed ADJ cana-1326 137 39 ball ball NOUN cana-1326 137 40 in in ADP cana-1326 137 41 complex complex ADJ cana-1326 137 42 valued value VERB cana-1326 137 43 b b NOUN cana-1326 137 44 - - PUNCT cana-1326 137 45 metric metric ADJ cana-1326 137 46 space space NOUN cana-1326 137 47 . . PUNCT cana-1326 138 1 corollary corollary ADJ cana-1326 138 2 1.5 1.5 NUM cana-1326 138 3 : : PUNCT cana-1326 138 4 let let VERB cana-1326 138 5 ( ( PUNCT cana-1326 138 6 x x NOUN cana-1326 138 7 , , PUNCT cana-1326 138 8 d d NOUN cana-1326 138 9 ) ) PUNCT cana-1326 138 10 be be AUX cana-1326 138 11 a a DET cana-1326 138 12 complete complete ADJ cana-1326 138 13 bicomplex bicomplex NOUN cana-1326 138 14 valued value VERB cana-1326 138 15 metric metric ADJ cana-1326 138 16 space space NOUN cana-1326 138 17 with with ADP cana-1326 138 18 coefficient coefficient NOUN cana-1326 138 19 sβ‰₯ sβ‰₯ PROPN cana-1326 138 20 1 1 NUM cana-1326 138 21 and and CCONJ cana-1326 138 22 π‘₯0 π‘₯0 PROPN cana-1326 138 23 ∈ ∈ PROPN cana-1326 138 24 𝑋. 𝑋. PROPN cana-1326 138 25 0≲ 0≲ PUNCT cana-1326 138 26 π‘Ÿ π‘Ÿ DET cana-1326 138 27 ∈ ∈ PROPN cana-1326 138 28 β„‚ β„‚ PROPN cana-1326 138 29 and and CCONJ cana-1326 138 30 a a DET cana-1326 138 31 , , PUNCT cana-1326 138 32 b b NOUN cana-1326 138 33 , , PUNCT cana-1326 138 34 c c NOUN cana-1326 138 35 , , PUNCT cana-1326 138 36 d d NOUN cana-1326 138 37 and and CCONJ cana-1326 138 38 e e NOUN cana-1326 138 39 are be AUX cana-1326 138 40 non non ADJ cana-1326 138 41 negative negative ADJ cana-1326 138 42 reals real NOUN cana-1326 138 43 such such ADJ cana-1326 138 44 that that SCONJ cana-1326 138 45 a+√2𝐡 a+√2𝐡 PROPN cana-1326 139 1 + + CCONJ cana-1326 139 2 √2𝐢 √2𝐢 PRON cana-1326 139 3 + + NOUN cana-1326 139 4 √2𝑠𝐷 √2𝑠𝐷 NOUN cana-1326 139 5 + + CCONJ cana-1326 139 6 √2𝑠𝐸 √2𝑠𝐸 PROPN cana-1326 139 7 < < X cana-1326 139 8 1 1 X cana-1326 139 9 . . PUNCT cana-1326 140 1 let let VERB cana-1326 140 2 t t PROPN cana-1326 140 3 : : PUNCT cana-1326 140 4 xβ†’x xβ†’x NUM cana-1326 140 5 are be AUX cana-1326 140 6 mapping map VERB cana-1326 140 7 satisfying satisfy VERB cana-1326 140 8 d(𝑇𝑛x, d(𝑇𝑛x, X cana-1326 140 9 οΏ½ οΏ½ NOUN cana-1326 140 10 𝑇𝑛y)≲𝑖2ad(x 𝑇𝑛y)≲𝑖2ad(x NOUN cana-1326 140 11 , , PUNCT cana-1326 140 12 y)+b y)+b PROPN cana-1326 140 13 𝑑(π‘₯,𝑇𝑛π‘₯)𝑑(𝑦,𝑇𝑛𝑦 𝑑(π‘₯,𝑇𝑛π‘₯)𝑑(𝑦,𝑇𝑛𝑦 NOUN cana-1326 140 14 ) ) PUNCT cana-1326 140 15 1+𝑑(π‘₯,𝑦 1+𝑑(π‘₯,𝑦 NOUN cana-1326 140 16 ) ) PUNCT cana-1326 141 1 + + CCONJ cana-1326 141 2 𝐢 𝐢 PROPN cana-1326 141 3 𝑑(𝑦,𝑇𝑛π‘₯)𝑑(π‘₯,𝑇𝑛𝑦 𝑑(𝑦,𝑇𝑛π‘₯)𝑑(π‘₯,𝑇𝑛𝑦 PROPN cana-1326 141 4 ) ) PUNCT cana-1326 141 5 1+𝑑(π‘₯,𝑦 1+𝑑(π‘₯,𝑦 NOUN cana-1326 141 6 ) ) PUNCT cana-1326 141 7 οΏ½ οΏ½ PROPN cana-1326 141 8 οΏ½ οΏ½ PROPN cana-1326 141 9 οΏ½ οΏ½ PROPN cana-1326 141 10 οΏ½ οΏ½ PROPN cana-1326 141 11 οΏ½ οΏ½ PROPN cana-1326 141 12 οΏ½ οΏ½ PROPN cana-1326 141 13 οΏ½ οΏ½ PROPN cana-1326 141 14 οΏ½ οΏ½ PROPN cana-1326 141 15 οΏ½ οΏ½ PROPN cana-1326 141 16 οΏ½ οΏ½ PROPN cana-1326 141 17 οΏ½ οΏ½ PROPN cana-1326 141 18 οΏ½ οΏ½ PROPN cana-1326 141 19 οΏ½ οΏ½ PROPN cana-1326 141 20 οΏ½ οΏ½ PROPN cana-1326 141 21 οΏ½ οΏ½ PROPN cana-1326 141 22 οΏ½ οΏ½ PROPN cana-1326 141 23 οΏ½ οΏ½ PROPN cana-1326 141 24 οΏ½ οΏ½ PROPN cana-1326 141 25 οΏ½ οΏ½ PROPN cana-1326 141 26 οΏ½ οΏ½ PROPN cana-1326 141 27 οΏ½ οΏ½ PROPN cana-1326 141 28 οΏ½ οΏ½ PROPN cana-1326 141 29 οΏ½ οΏ½ PROPN cana-1326 141 30 οΏ½ οΏ½ PROPN cana-1326 141 31 οΏ½ οΏ½ PROPN cana-1326 141 32 οΏ½ οΏ½ PROPN cana-1326 141 33 οΏ½ οΏ½ PROPN cana-1326 141 34 οΏ½ οΏ½ PROPN cana-1326 141 35 οΏ½ οΏ½ PROPN cana-1326 141 36 οΏ½ οΏ½ PROPN cana-1326 141 37 οΏ½ οΏ½ PROPN cana-1326 141 38 οΏ½ οΏ½ PROPN cana-1326 141 39 οΏ½ οΏ½ PROPN cana-1326 141 40 οΏ½ οΏ½ PROPN cana-1326 141 41 οΏ½ οΏ½ PROPN cana-1326 141 42 οΏ½ οΏ½ PROPN cana-1326 141 43 οΏ½ οΏ½ PROPN cana-1326 141 44 οΏ½ οΏ½ PROPN cana-1326 141 45 οΏ½ οΏ½ PROPN cana-1326 141 46 οΏ½ οΏ½ PROPN cana-1326 141 47 οΏ½ οΏ½ PROPN cana-1326 141 48 οΏ½ οΏ½ PROPN cana-1326 141 49 οΏ½ οΏ½ PROPN cana-1326 141 50 οΏ½ οΏ½ PROPN cana-1326 141 51 οΏ½ οΏ½ PROPN cana-1326 141 52 οΏ½ οΏ½ PROPN cana-1326 141 53 οΏ½ οΏ½ PROPN cana-1326 141 54 οΏ½ οΏ½ PROPN cana-1326 141 55 οΏ½ οΏ½ PROPN cana-1326 141 56 οΏ½ οΏ½ PROPN cana-1326 141 57 οΏ½ οΏ½ PROPN cana-1326 141 58 οΏ½ οΏ½ PROPN cana-1326 141 59 οΏ½ οΏ½ PROPN cana-1326 141 60 οΏ½ οΏ½ PROPN cana-1326 141 61 οΏ½ οΏ½ PROPN cana-1326 141 62 οΏ½ οΏ½ PROPN cana-1326 141 63 οΏ½ οΏ½ PROPN cana-1326 141 64 οΏ½ οΏ½ PROPN cana-1326 141 65 οΏ½ οΏ½ PROPN cana-1326 141 66 οΏ½ οΏ½ PROPN cana-1326 141 67 οΏ½ οΏ½ PROPN cana-1326 141 68 οΏ½ οΏ½ PROPN cana-1326 141 69 οΏ½ οΏ½ PROPN cana-1326 141 70 οΏ½ οΏ½ PROPN cana-1326 141 71 οΏ½ οΏ½ PROPN cana-1326 141 72 οΏ½ οΏ½ PROPN cana-1326 141 73 οΏ½ οΏ½ PROPN cana-1326 141 74 οΏ½ οΏ½ PROPN cana-1326 141 75 οΏ½ οΏ½ PROPN cana-1326 141 76 οΏ½ οΏ½ PROPN cana-1326 141 77 οΏ½ οΏ½ PROPN cana-1326 141 78 οΏ½ οΏ½ PROPN cana-1326 141 79 οΏ½ οΏ½ PROPN cana-1326 141 80 οΏ½ οΏ½ PROPN cana-1326 141 81 οΏ½ οΏ½ PROPN cana-1326 141 82 οΏ½ οΏ½ PROPN cana-1326 141 83 οΏ½ οΏ½ PROPN cana-1326 141 84 οΏ½ οΏ½ PROPN cana-1326 141 85 οΏ½ οΏ½ PROPN cana-1326 141 86 οΏ½ οΏ½ PROPN cana-1326 141 87 οΏ½ οΏ½ PROPN cana-1326 141 88 οΏ½ οΏ½ PROPN cana-1326 141 89 οΏ½ οΏ½ PROPN cana-1326 141 90 οΏ½ οΏ½ PROPN cana-1326 141 91 + + PROPN cana-1326 141 92 𝐷 𝐷 PROPN cana-1326 141 93 𝑑(π‘₯,𝑇𝑛π‘₯)𝑑(π‘₯,𝑇𝑛𝑦 𝑑(π‘₯,𝑇𝑛π‘₯)𝑑(π‘₯,𝑇𝑛𝑦 NOUN cana-1326 141 94 ) ) PUNCT cana-1326 141 95 1+𝑑(π‘₯,𝑦 1+𝑑(π‘₯,𝑦 NOUN cana-1326 141 96 ) ) PUNCT cana-1326 141 97 οΏ½ οΏ½ PROPN cana-1326 141 98 οΏ½ οΏ½ PROPN cana-1326 141 99 + + NUM cana-1326 141 100 𝐸 𝐸 PROPN cana-1326 141 101 𝑑(𝑦,𝑇𝑛π‘₯)𝑑(𝑦,𝑇𝑛𝑦 𝑑(𝑦,𝑇𝑛π‘₯)𝑑(𝑦,𝑇𝑛𝑦 PROPN cana-1326 141 102 ) ) PUNCT cana-1326 141 103 1+𝑑(π‘₯,𝑦 1+𝑑(π‘₯,𝑦 NOUN cana-1326 141 104 ) ) PUNCT cana-1326 141 105 for for ADP cana-1326 141 106 all all DET cana-1326 141 107 x x ADJ cana-1326 141 108 , , PUNCT cana-1326 141 109 y∈ y∈ PROPN cana-1326 141 110 b(π‘₯0 b(π‘₯0 NOUN cana-1326 141 111 , , PUNCT cana-1326 141 112 π‘Ÿ)Μ…Μ… π‘Ÿ)Μ…Μ… PROPN cana-1326 141 113 Μ…Μ… Μ…Μ… PROPN cana-1326 141 114 Μ…Μ… Μ…Μ… PROPN cana-1326 141 115 Μ…Μ… Μ…Μ… PROPN cana-1326 141 116 Μ…Μ… Μ…Μ… PROPN cana-1326 141 117 . . PUNCT cana-1326 142 1 if if SCONJ cana-1326 142 2 ‖𝑑(π‘₯0 ‖𝑑(π‘₯0 PROPN cana-1326 142 3 , , PUNCT cana-1326 142 4 𝑇 𝑇 PROPN cana-1326 142 5 𝑛π‘₯0β€– 𝑛π‘₯0β€– PUNCT cana-1326 142 6 ≲𝑖2 ≲𝑖2 NOUN cana-1326 142 7 ( ( PUNCT cana-1326 142 8 1 1 NUM cana-1326 142 9 βˆ’ βˆ’ PROPN cana-1326 142 10 πœ†)|π‘Ÿ| πœ†)|π‘Ÿ| PROPN cana-1326 142 11 where where SCONJ cana-1326 142 12 Ξ»= Ξ»= VERB cana-1326 142 13 max max PROPN cana-1326 142 14 { { PUNCT cana-1326 142 15 𝐴+√2𝑠𝐷 𝐴+√2𝑠𝐷 PROPN cana-1326 142 16 1βˆ’π΅βˆ’βˆš2𝑠𝐷 1βˆ’π΅βˆ’βˆš2𝑠𝐷 NUM cana-1326 142 17 , , PUNCT cana-1326 142 18 𝐴+√2𝑠𝐸 𝐴+√2𝑠𝐸 PROPN cana-1326 142 19 1βˆ’βˆš2π΅βˆ’βˆš2𝑠𝐸 1βˆ’βˆš2π΅βˆ’βˆš2𝑠𝐸 NUM cana-1326 142 20 + + ADJ cana-1326 142 21 , , PUNCT cana-1326 142 22 then then ADV cana-1326 142 23 there there PRON cana-1326 142 24 exist exist VERB cana-1326 142 25 a a DET cana-1326 142 26 unique unique ADJ cana-1326 142 27 point point NOUN cana-1326 142 28 u∈ u∈ NOUN cana-1326 142 29 b(π‘₯0 b(π‘₯0 NOUN cana-1326 142 30 , , PUNCT cana-1326 142 31 π‘Ÿ)Μ…Μ… π‘Ÿ)Μ…Μ… PROPN cana-1326 142 32 Μ…Μ… Μ…Μ… PROPN cana-1326 142 33 Μ…Μ… Μ…Μ… PROPN cana-1326 142 34 Μ…Μ… Μ…Μ… PROPN cana-1326 142 35 Μ…Μ… Μ…Μ… PROPN cana-1326 142 36 such such ADJ cana-1326 142 37 that that SCONJ cana-1326 142 38 u u PROPN cana-1326 142 39 = = PROPN cana-1326 142 40 tu tu PROPN cana-1326 142 41 . . PUNCT cana-1326 142 42 proof proof NOUN cana-1326 142 43 : : PUNCT cana-1326 142 44 for for ADP cana-1326 142 45 some some DET cana-1326 142 46 fixed fix VERB cana-1326 142 47 n n CCONJ cana-1326 142 48 , , PUNCT cana-1326 142 49 we we PRON cana-1326 142 50 obtain obtain VERB cana-1326 142 51 u∈ u∈ PROPN cana-1326 142 52 b(π‘₯0 b(π‘₯0 NOUN cana-1326 142 53 , , PUNCT cana-1326 142 54 π‘Ÿ)Μ…Μ… π‘Ÿ)Μ…Μ… PROPN cana-1326 142 55 Μ…Μ… Μ…Μ… PROPN cana-1326 142 56 Μ…Μ… Μ…Μ… PROPN cana-1326 142 57 Μ…Μ… Μ…Μ… PROPN cana-1326 142 58 Μ…Μ… Μ…Μ… PROPN cana-1326 142 59 οΏ½ οΏ½ PROPN cana-1326 143 1 such such ADJ cana-1326 143 2 that that SCONJ cana-1326 143 3 𝑇𝑛𝑒 𝑇𝑛𝑒 PROPN cana-1326 143 4 = = PUNCT cana-1326 143 5 𝑒. 𝑒. NOUN cana-1326 143 6 the the DET cana-1326 143 7 uniqueness uniqueness NOUN cana-1326 143 8 follows follow VERB cana-1326 143 9 from from ADP cana-1326 143 10 d(tu d(tu PROPN cana-1326 143 11 , , PUNCT cana-1326 143 12 u)= u)= NOUN cana-1326 143 13 οΏ½ οΏ½ NOUN cana-1326 143 14 𝑑(𝑇𝑇𝑛𝑒 𝑑(𝑇𝑇𝑛𝑒 NOUN cana-1326 143 15 , , PUNCT cana-1326 143 16 𝑇𝑛𝑒 𝑇𝑛𝑒 PROPN cana-1326 143 17 ) ) PUNCT cana-1326 143 18 ≲𝑖2a ≲𝑖2a PROPN cana-1326 143 19 οΏ½ οΏ½ NOUN cana-1326 143 20 𝑑(𝑇𝑒 𝑑(𝑇𝑒 NOUN cana-1326 143 21 , , PUNCT cana-1326 143 22 𝑒 𝑒 X cana-1326 143 23 ) ) PUNCT cana-1326 143 24 + + CCONJ cana-1326 143 25 οΏ½ οΏ½ PROPN cana-1326 143 26 𝐡 𝐡 NOUN cana-1326 143 27 𝑑(𝑇𝑒,𝑇𝑛𝑇𝑒)𝑑(𝑒,𝑇𝑛𝑒 𝑑(𝑇𝑒,𝑇𝑛𝑇𝑒)𝑑(𝑒,𝑇𝑛𝑒 NOUN cana-1326 143 28 ) ) PUNCT cana-1326 143 29 1+𝑑(𝑇𝑒,𝑒 1+𝑑(𝑇𝑒,𝑒 NUM cana-1326 143 30 ) ) PUNCT cana-1326 143 31 + + CCONJ cana-1326 143 32 𝐢 𝐢 PROPN cana-1326 143 33 𝑑(𝑒,𝑇𝑛𝑇𝑒)𝑑(𝑇𝑒,𝑇𝑛𝑒 𝑑(𝑒,𝑇𝑛𝑇𝑒)𝑑(𝑇𝑒,𝑇𝑛𝑒 NOUN cana-1326 143 34 ) ) PUNCT cana-1326 143 35 1+𝑑(𝑇𝑒,𝑒 1+𝑑(𝑇𝑒,𝑒 NUM cana-1326 143 36 ) ) PUNCT cana-1326 143 37 + + ADJ cana-1326 143 38 𝐷 𝐷 NOUN cana-1326 143 39 𝑑(𝑇𝑒,𝑇𝑛𝑇𝑒)𝑑(𝑇𝑒,𝑇𝑛𝑒 𝑑(𝑇𝑒,𝑇𝑛𝑇𝑒)𝑑(𝑇𝑒,𝑇𝑛𝑒 NOUN cana-1326 143 40 ) ) PUNCT cana-1326 143 41 1+𝑑(𝑇𝑒,𝑒 1+𝑑(𝑇𝑒,𝑒 NUM cana-1326 143 42 ) ) PUNCT cana-1326 143 43 + + CCONJ cana-1326 143 44 𝐸 𝐸 PROPN cana-1326 143 45 𝑑(𝑒,𝑇𝑛𝑇𝑒)𝑑(𝑒,𝑇𝑛𝑒 𝑑(𝑒,𝑇𝑛𝑇𝑒)𝑑(𝑒,𝑇𝑛𝑒 PROPN cana-1326 143 46 ) ) PUNCT cana-1326 143 47 1+𝑑(𝑇𝑒,𝑒 1+𝑑(𝑇𝑒,𝑒 NUM cana-1326 143 48 ) ) PUNCT cana-1326 143 49 ≲𝑖2a ≲𝑖2a ADJ cana-1326 143 50 οΏ½ οΏ½ NOUN cana-1326 143 51 𝑑(𝑇𝑒 𝑑(𝑇𝑒 NOUN cana-1326 143 52 , , PUNCT cana-1326 143 53 𝑒 𝑒 X cana-1326 143 54 ) ) PUNCT cana-1326 143 55 + + CCONJ cana-1326 143 56 𝐢 𝐢 PROPN cana-1326 143 57 𝑑(𝑒,𝑇𝑛𝑇𝑒)𝑑(𝑇𝑒,𝑇𝑛𝑒 𝑑(𝑒,𝑇𝑛𝑇𝑒)𝑑(𝑇𝑒,𝑇𝑛𝑒 NOUN cana-1326 143 58 ) ) PUNCT cana-1326 143 59 1+𝑑(𝑇𝑒,𝑒 1+𝑑(𝑇𝑒,𝑒 NUM cana-1326 143 60 ) ) PUNCT cana-1326 143 61 + + CCONJ cana-1326 143 62 𝐷 𝐷 PROPN cana-1326 143 63 𝑑(𝑇𝑒,𝑇𝑛𝑇𝑒)𝑑(𝑇𝑒,𝑇𝑛𝑒 𝑑(𝑇𝑒,𝑇𝑛𝑇𝑒)𝑑(𝑇𝑒,𝑇𝑛𝑒 NOUN cana-1326 143 64 ) ) PUNCT cana-1326 143 65 1+𝑑(𝑇𝑒,𝑒 1+𝑑(𝑇𝑒,𝑒 NUM cana-1326 143 66 ) ) PUNCT cana-1326 143 67 ≲𝑖2a ≲𝑖2a ADJ cana-1326 143 68 οΏ½ οΏ½ NOUN cana-1326 143 69 𝑑(𝑇𝑒 𝑑(𝑇𝑒 NOUN cana-1326 143 70 , , PUNCT cana-1326 143 71 𝑒 𝑒 X cana-1326 143 72 ) ) PUNCT cana-1326 143 73 + + CCONJ cana-1326 143 74 𝐢 𝐢 PROPN cana-1326 143 75 𝑑(𝑒,𝑇𝑒)𝑑(𝑇𝑒,𝑒 𝑑(𝑒,𝑇𝑒)𝑑(𝑇𝑒,𝑒 NOUN cana-1326 143 76 ) ) PUNCT cana-1326 143 77 1+𝑑(𝑇𝑒,𝑒 1+𝑑(𝑇𝑒,𝑒 NOUN cana-1326 143 78 ) ) PUNCT cana-1326 143 79 taking take VERB cana-1326 143 80 norm norm NOUN cana-1326 143 81 in in ADP cana-1326 143 82 above above ADV cana-1326 143 83 , , PUNCT cana-1326 143 84 we we PRON cana-1326 143 85 get get VERB cana-1326 143 86 ‖𝑑(𝑇𝑒 ‖𝑑(𝑇𝑒 ADJ cana-1326 143 87 , , PUNCT cana-1326 143 88 𝑒)β€– 𝑒)β€– ADV cana-1326 143 89 ≀ ≀ PROPN cana-1326 143 90 a‖𝑑(𝑇𝑒 a‖𝑑(𝑇𝑒 PROPN cana-1326 143 91 , , PUNCT cana-1326 143 92 𝑒)β€– 𝑒)β€– X cana-1326 144 1 + + CCONJ cana-1326 144 2 𝐢√2 𝐢√2 NOUN cana-1326 144 3 ‖𝑑(𝑒,𝑇𝑒)‖‖𝑑(𝑇𝑒,𝑒)β€– ‖𝑑(𝑒,𝑇𝑒)‖‖𝑑(𝑇𝑒,𝑒)β€– VERB cana-1326 144 4 β€–1+𝑑(𝑇𝑒,𝑒)β€– β€–1+𝑑(𝑇𝑒,𝑒)β€– PUNCT cana-1326 144 5 ≀ ≀ PROPN cana-1326 144 6 𝐴‖𝑑(𝑇𝑒 𝐴‖𝑑(𝑇𝑒 NOUN cana-1326 144 7 , , PUNCT cana-1326 144 8 𝑒)β€– 𝑒)β€– X cana-1326 144 9 + + CCONJ cana-1326 144 10 𝐢√2‖𝑑(𝑇𝑒 𝐢√2‖𝑑(𝑇𝑒 ADJ cana-1326 144 11 , , PUNCT cana-1326 144 12 𝑒)β€– 𝑒)β€– NOUN cana-1326 145 1 [ [ X cana-1326 145 2 as as ADP cana-1326 145 3 β€–1 β€–1 NOUN cana-1326 145 4 + + CCONJ cana-1326 145 5 𝑑(𝑒 𝑑(𝑒 ADJ cana-1326 145 6 , , PUNCT cana-1326 145 7 𝑒)β€– 𝑒)β€– X cana-1326 145 8 > > X cana-1326 145 9 ‖𝑑(𝑒 ‖𝑑(𝑒 X cana-1326 145 10 , , PUNCT cana-1326 145 11 𝑒)β€– 𝑒)β€– X cana-1326 145 12 ] ] PUNCT cana-1326 145 13 ∴ ∴ PROPN cana-1326 145 14 ‖𝑑(𝑇𝑒 ‖𝑑(𝑇𝑒 NOUN cana-1326 145 15 , , PUNCT cana-1326 145 16 𝑒)β€– 𝑒)β€– ADV cana-1326 145 17 ≀ ≀ X cana-1326 145 18 ( ( PUNCT cana-1326 145 19 𝐴 𝐴 PROPN cana-1326 145 20 + + CCONJ cana-1326 145 21 𝐢√2)‖𝑑(𝑇𝑒 𝐢√2)‖𝑑(𝑇𝑒 PROPN cana-1326 145 22 , , PUNCT cana-1326 145 23 𝑒)β€– 𝑒)β€– ADV cana-1326 145 24 this this PRON cana-1326 145 25 is be AUX cana-1326 145 26 a a DET cana-1326 145 27 contradiction contradiction NOUN cana-1326 145 28 . . PUNCT cana-1326 146 1 so so ADV cana-1326 146 2 u=𝑇𝑛𝑒=tu u=𝑇𝑛𝑒=tu PROPN cana-1326 146 3 . . PUNCT cana-1326 147 1 therefore therefore ADV cana-1326 147 2 the the DET cana-1326 147 3 fixed fixed ADJ cana-1326 147 4 point point NOUN cana-1326 147 5 of of ADP cana-1326 147 6 t t PROPN cana-1326 147 7 is be AUX cana-1326 147 8 unique unique ADJ cana-1326 147 9 . . PUNCT cana-1326 148 1 theorem theorem NOUN cana-1326 148 2 2 2 NUM cana-1326 148 3 : : PUNCT cana-1326 148 4 let let VERB cana-1326 148 5 ( ( PUNCT cana-1326 148 6 x x NOUN cana-1326 148 7 , , PUNCT cana-1326 148 8 d d NOUN cana-1326 148 9 ) ) PUNCT cana-1326 148 10 be be AUX cana-1326 148 11 a a DET cana-1326 148 12 bi bi ADJ cana-1326 148 13 - - ADJ cana-1326 148 14 complex complex ADJ cana-1326 148 15 valued value VERB cana-1326 148 16 complete complete ADJ cana-1326 148 17 metric metric ADJ cana-1326 148 18 space space NOUN cana-1326 148 19 and and CCONJ cana-1326 148 20 t t PROPN cana-1326 148 21 , , PUNCT cana-1326 148 22 s s PART cana-1326 148 23 : : PUNCT cana-1326 149 1 xβ†’x xβ†’x NUM cana-1326 149 2 be be AUX cana-1326 149 3 a a DET cana-1326 149 4 self self NOUN cana-1326 149 5 map map NOUN cana-1326 149 6 satisfying satisfy VERB cana-1326 149 7 the the DET cana-1326 149 8 following follow VERB cana-1326 149 9 conditions condition NOUN cana-1326 149 10 d(s(x),t(y d(s(x),t(y PROPN cana-1326 149 11 ) ) PUNCT cana-1326 149 12 ) ) PUNCT cana-1326 150 1 ≲𝑖2 ≲𝑖2 PROPN cana-1326 150 2 𝛼 𝛼 X cana-1326 150 3 οΏ½ οΏ½ NOUN cana-1326 150 4 max[d(x max[d(x NOUN cana-1326 150 5 , , PUNCT cana-1326 150 6 y y PROPN cana-1326 150 7 ) ) PUNCT cana-1326 150 8 , , PUNCT cana-1326 150 9 𝑑(π‘₯,𝑆(π‘₯))𝑑(𝑦,𝑇(𝑦 𝑑(π‘₯,𝑆(π‘₯))𝑑(𝑦,𝑇(𝑦 NOUN cana-1326 150 10 ) ) PUNCT cana-1326 150 11 ) ) PUNCT cana-1326 151 1 1+d(sx 1+d(sx NUM cana-1326 151 2 , , PUNCT cana-1326 151 3 ty ty NOUN cana-1326 151 4 ) ) PUNCT cana-1326 151 5 οΏ½ οΏ½ PROPN cana-1326 151 6 ] ] PUNCT cana-1326 151 7 … … PUNCT cana-1326 151 8 . . PUNCT cana-1326 152 1 ( ( PUNCT cana-1326 152 2 2.1 2.1 NUM cana-1326 152 3 ) ) PUNCT cana-1326 152 4 for for ADP cana-1326 152 5 all all DET cana-1326 152 6 x x NOUN cana-1326 152 7 , , PUNCT cana-1326 152 8 y∈x y∈x NOUN cana-1326 152 9 , , PUNCT cana-1326 152 10 where where SCONJ cana-1326 152 11 Ξ± Ξ± NOUN cana-1326 152 12 is be AUX cana-1326 152 13 a a DET cana-1326 152 14 real real NOUN cana-1326 152 15 with with ADP cana-1326 152 16 0 0 NUM cana-1326 152 17 < < X cana-1326 152 18 𝛼 𝛼 X cana-1326 152 19 < < X cana-1326 152 20 1 1 NUM cana-1326 152 21 . . PUNCT cana-1326 153 1 then then ADV cana-1326 153 2 s s VERB cana-1326 153 3 and and CCONJ cana-1326 153 4 t t PROPN cana-1326 153 5 have have VERB cana-1326 153 6 a a DET cana-1326 153 7 unique unique ADJ cana-1326 153 8 common common ADJ cana-1326 153 9 fixed fix VERB cana-1326 153 10 point point NOUN cana-1326 153 11 . . PUNCT cana-1326 154 1 proof proof NOUN cana-1326 154 2 : : PUNCT cana-1326 154 3 let let VERB cana-1326 154 4 x∈x x∈x PROPN cana-1326 154 5 be be AUX cana-1326 154 6 arbitrary arbitrary ADJ cana-1326 154 7 . . PUNCT cana-1326 155 1 we we PRON cana-1326 155 2 define define VERB cana-1326 155 3 a a DET cana-1326 155 4 sequence sequence NOUN cana-1326 155 5 { { PUNCT cana-1326 155 6 π‘₯𝑛+ π‘₯𝑛+ VERB cana-1326 155 7 in in ADP cana-1326 155 8 x x PUNCT cana-1326 155 9 as as SCONJ cana-1326 155 10 follows follow VERB cana-1326 155 11 π‘₯2π‘˜+1 π‘₯2π‘˜+1 NOUN cana-1326 155 12 = = NOUN cana-1326 155 13 s(π‘₯2π‘˜ s(π‘₯2π‘˜ X cana-1326 155 14 ) ) PUNCT cana-1326 155 15 and and CCONJ cana-1326 155 16 π‘₯2π‘˜+2 π‘₯2π‘˜+2 NOUN cana-1326 155 17 = = NOUN cana-1326 155 18 t(π‘₯2π‘˜+1 t(π‘₯2π‘˜+1 NOUN cana-1326 155 19 ) ) PUNCT cana-1326 155 20 for for ADP cana-1326 155 21 k=0,1,2 k=0,1,2 PROPN cana-1326 155 22 … … X cana-1326 155 23 . . PUNCT cana-1326 156 1 then then ADV cana-1326 156 2 d(x2k+1 d(x2k+1 NOUN cana-1326 156 3 , , PUNCT cana-1326 156 4 x2k+2 x2k+2 NUM cana-1326 156 5 ) ) PUNCT cana-1326 156 6 = = SYM cana-1326 156 7 d(s(π‘₯2π‘˜ d(s(π‘₯2π‘˜ PROPN cana-1326 156 8 ) ) PUNCT cana-1326 156 9 , , PUNCT cana-1326 156 10 t(π‘₯2π‘˜+1 t(π‘₯2π‘˜+1 NOUN cana-1326 156 11 ) ) PUNCT cana-1326 156 12 ) ) PUNCT cana-1326 157 1 communications communication NOUN cana-1326 157 2 on on ADP cana-1326 157 3 applied apply VERB cana-1326 157 4 nonlinear nonlinear ADJ cana-1326 157 5 analysis analysis NOUN cana-1326 157 6 issn issn NOUN cana-1326 157 7 : : PUNCT cana-1326 157 8 1074 1074 NUM cana-1326 157 9 - - PUNCT cana-1326 157 10 133x 133x NUM cana-1326 157 11 vol vol NOUN cana-1326 157 12 31 31 NUM cana-1326 157 13 no no NOUN cana-1326 157 14 . . PUNCT cana-1326 158 1 7s 7 NOUN cana-1326 158 2 ( ( PUNCT cana-1326 158 3 2024 2024 NUM cana-1326 158 4 ) ) PUNCT cana-1326 158 5 473 473 NUM cana-1326 158 6 https://internationalpubls.com https://internationalpubls.com X cana-1326 158 7 ≲𝑖2 ≲𝑖2 ADJ cana-1326 158 8 𝛼max 𝛼max X cana-1326 158 9 οΏ½ οΏ½ NOUN cana-1326 158 10 ,𝑑(π‘₯2π‘˜ ,𝑑(π‘₯2π‘˜ PUNCT cana-1326 158 11 , , PUNCT cana-1326 158 12 x2k+1 x2k+1 PROPN cana-1326 158 13 ) ) PUNCT cana-1326 158 14 , , PUNCT cana-1326 158 15 𝑑(π‘₯2π‘˜,s(π‘₯2π‘˜))𝑑(x2k+1,t(π‘₯2π‘˜+1 𝑑(π‘₯2π‘˜,s(π‘₯2π‘˜))𝑑(x2k+1,t(π‘₯2π‘˜+1 NOUN cana-1326 158 16 ) ) PUNCT cana-1326 158 17 ) ) PUNCT cana-1326 158 18 1+𝑑(𝑆π‘₯2π‘˜,tx2k+1 1+𝑑(𝑆π‘₯2π‘˜,tx2k+1 X cana-1326 158 19 ) ) PUNCT cana-1326 158 20 ∴ ∴ NOUN cana-1326 158 21 d(x2k+1 d(x2k+1 NOUN cana-1326 158 22 , , PUNCT cana-1326 158 23 x2k+2 x2k+2 NOUN cana-1326 158 24 ) ) PUNCT cana-1326 158 25 ≲𝑖2 ≲𝑖2 ADJ cana-1326 158 26 𝛼d(x2k 𝛼d(x2k NOUN cana-1326 158 27 , , PUNCT cana-1326 158 28 x2k+1 x2k+1 NOUN cana-1326 158 29 ) ) PUNCT cana-1326 158 30 ( ( PUNCT cana-1326 158 31 2.2 2.2 NUM cana-1326 158 32 ) ) PUNCT cana-1326 158 33 similarly similarly ADV cana-1326 158 34 , , PUNCT cana-1326 158 35 d(x2k+2 d(x2k+2 NOUN cana-1326 158 36 , , PUNCT cana-1326 158 37 x2k+3 x2k+3 NUM cana-1326 158 38 ) ) PUNCT cana-1326 158 39 = = PUNCT cana-1326 159 1 d d X cana-1326 159 2 ( ( PUNCT cana-1326 159 3 t(π‘₯2π‘˜+1 t(π‘₯2π‘˜+1 NOUN cana-1326 159 4 ) ) PUNCT cana-1326 159 5 , , PUNCT cana-1326 159 6 s(π‘₯2π‘˜+2 s(π‘₯2π‘˜+2 NUM cana-1326 159 7 ) ) PUNCT cana-1326 159 8 ) ) PUNCT cana-1326 160 1 = = SYM cana-1326 160 2 d(s(π‘₯2π‘˜+2 d(s(π‘₯2π‘˜+2 NOUN cana-1326 160 3 ) ) PUNCT cana-1326 160 4 , , PUNCT cana-1326 160 5 t(π‘₯2π‘˜+1 t(π‘₯2π‘˜+1 NOUN cana-1326 160 6 ) ) PUNCT cana-1326 160 7 ) ) PUNCT cana-1326 161 1 ≲𝑖2 ≲𝑖2 PROPN cana-1326 161 2 οΏ½ οΏ½ PROPN cana-1326 161 3 𝛼max 𝛼max X cana-1326 161 4 οΏ½ οΏ½ NOUN cana-1326 161 5 ,d(π‘₯2π‘˜+2 ,d(π‘₯2π‘˜+2 PUNCT cana-1326 161 6 , , PUNCT cana-1326 161 7 x2k+1 x2k+1 PROPN cana-1326 161 8 ) ) PUNCT cana-1326 161 9 , , PUNCT cana-1326 161 10 𝑑(π‘₯2π‘˜+2,s(π‘₯2π‘˜+2))𝑑(x2k+1,t(π‘₯2π‘˜+1 𝑑(π‘₯2π‘˜+2,s(π‘₯2π‘˜+2))𝑑(x2k+1,t(π‘₯2π‘˜+1 NOUN cana-1326 161 11 ) ) PUNCT cana-1326 161 12 ) ) PUNCT cana-1326 161 13 1+𝑑(𝑆π‘₯2π‘˜+2,𝑇π‘₯2π‘˜+1 1+𝑑(𝑆π‘₯2π‘˜+2,𝑇π‘₯2π‘˜+1 NOUN cana-1326 161 14 ) ) PUNCT cana-1326 161 15 ] ] PUNCT cana-1326 162 1 ≲𝑖2 ≲𝑖2 PROPN cana-1326 162 2 οΏ½ οΏ½ PROPN cana-1326 162 3 𝛼d(x2k+2 𝛼d(x2k+2 NUM cana-1326 162 4 , , PUNCT cana-1326 162 5 x2k+1)= x2k+1)= PRON cana-1326 162 6 𝛼d(x2k+1 𝛼d(x2k+1 PROPN cana-1326 162 7 , , PUNCT cana-1326 162 8 x2k+2 x2k+2 PROPN cana-1326 162 9 ) ) PUNCT cana-1326 162 10 ( ( PUNCT cana-1326 162 11 2.3 2.3 NUM cana-1326 162 12 ) ) PUNCT cana-1326 162 13 then then ADV cana-1326 162 14 from from ADP cana-1326 162 15 ( ( PUNCT cana-1326 162 16 2.2 2.2 NUM cana-1326 162 17 ) ) PUNCT cana-1326 162 18 and and CCONJ cana-1326 162 19 ( ( PUNCT cana-1326 162 20 2.3 2.3 NUM cana-1326 162 21 ) ) PUNCT cana-1326 162 22 , , PUNCT cana-1326 162 23 we we PRON cana-1326 162 24 get get VERB cana-1326 162 25 , , PUNCT cana-1326 162 26 d(π‘₯𝑛+1 d(π‘₯𝑛+1 PROPN cana-1326 162 27 , , PUNCT cana-1326 162 28 π‘₯𝑛+2 π‘₯𝑛+2 CCONJ cana-1326 162 29 ) ) PUNCT cana-1326 162 30 ≲𝑖2 ≲𝑖2 PROPN cana-1326 162 31 𝛼 𝛼 ADP cana-1326 162 32 d(π‘₯𝑛 d(π‘₯𝑛 PROPN cana-1326 162 33 , , PUNCT cana-1326 162 34 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-1326 162 35 ) ) PUNCT cana-1326 162 36 ≲𝑖2 ≲𝑖2 ADJ cana-1326 162 37 𝛼 𝛼 NOUN cana-1326 162 38 2𝑑(π‘₯π‘›βˆ’1 2𝑑(π‘₯π‘›βˆ’1 PROPN cana-1326 162 39 , , PUNCT cana-1326 162 40 π‘₯𝑛) π‘₯𝑛) PROPN cana-1326 162 41 … … SYM cana-1326 162 42 … … PUNCT cana-1326 162 43 … … PUNCT cana-1326 162 44 … … PUNCT cana-1326 162 45 … … PUNCT cana-1326 162 46 ≲𝑖2 ≲𝑖2 ADJ cana-1326 162 47 𝛼 𝛼 NOUN cana-1326 162 48 𝑛+1d(π‘₯0 𝑛+1d(π‘₯0 ADJ cana-1326 162 49 , , PUNCT cana-1326 162 50 π‘₯1 π‘₯1 NOUN cana-1326 162 51 ) ) PUNCT cana-1326 162 52 βˆ€π‘› βˆ€π‘› NOUN cana-1326 162 53 ∈ ∈ PROPN cana-1326 162 54 β„•. β„•. PROPN cana-1326 162 55 now now ADV cana-1326 162 56 for for ADP cana-1326 162 57 all all DET cana-1326 162 58 m m PROPN cana-1326 162 59 , , PUNCT cana-1326 162 60 nβˆˆβ„• nβˆˆβ„• NOUN cana-1326 162 61 we we PRON cana-1326 162 62 have have VERB cana-1326 162 63 , , PUNCT cana-1326 162 64 d(π‘₯𝑛 d(π‘₯𝑛 PROPN cana-1326 162 65 , , PUNCT cana-1326 162 66 π‘₯π‘š+𝑛 π‘₯π‘š+𝑛 PROPN cana-1326 162 67 ) ) PUNCT cana-1326 162 68 ≲𝑖2 ≲𝑖2 PROPN cana-1326 162 69 d(π‘₯𝑛 d(π‘₯𝑛 PROPN cana-1326 162 70 , , PUNCT cana-1326 162 71 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-1326 162 72 ) ) PUNCT cana-1326 162 73 + + CCONJ cana-1326 163 1 d(π‘₯𝑛+1 d(π‘₯𝑛+1 PROPN cana-1326 163 2 , , PUNCT cana-1326 163 3 π‘₯𝑛+2 π‘₯𝑛+2 X cana-1326 163 4 ) ) PUNCT cana-1326 164 1 + + NUM cana-1326 164 2 β‹―+ β‹―+ NUM cana-1326 164 3 d(π‘₯π‘š+π‘›βˆ’1 d(π‘₯π‘š+π‘›βˆ’1 NOUN cana-1326 164 4 , , PUNCT cana-1326 164 5 π‘₯π‘š+𝑛 π‘₯π‘š+𝑛 NOUN cana-1326 164 6 ) ) PUNCT cana-1326 164 7 ≲𝑖2 ≲𝑖2 PROPN cana-1326 164 8 𝛼 𝛼 NOUN cana-1326 164 9 𝑛d(π‘₯0 𝑛d(π‘₯0 NOUN cana-1326 164 10 , , PUNCT cana-1326 164 11 π‘₯1 π‘₯1 NOUN cana-1326 164 12 ) ) PUNCT cana-1326 164 13 + + SYM cana-1326 164 14 𝛼𝑛+1 𝛼𝑛+1 NUM cana-1326 164 15 οΏ½ οΏ½ PROPN cana-1326 164 16 d(π‘₯0 d(π‘₯0 ADJ cana-1326 164 17 , , PUNCT cana-1326 164 18 π‘₯1 π‘₯1 NOUN cana-1326 164 19 ) ) PUNCT cana-1326 164 20 + + NUM cana-1326 164 21 β‹―+ β‹―+ NUM cana-1326 164 22 π›Όπ‘š+π‘›βˆ’1d(π‘₯0 π›Όπ‘š+π‘›βˆ’1d(π‘₯0 ADJ cana-1326 164 23 , , PUNCT cana-1326 164 24 π‘₯1 π‘₯1 NOUN cana-1326 164 25 ) ) PUNCT cana-1326 164 26 ∴ ∴ PROPN cana-1326 164 27 d(π‘₯𝑛 d(π‘₯𝑛 PROPN cana-1326 164 28 , , PUNCT cana-1326 164 29 π‘₯π‘š+𝑛 π‘₯π‘š+𝑛 PROPN cana-1326 164 30 ) ) PUNCT cana-1326 164 31 ≲𝑖2 ≲𝑖2 PROPN cana-1326 164 32 𝛼 𝛼 ADP cana-1326 164 33 𝑛(1 𝑛(1 PROPN cana-1326 164 34 + + CCONJ cana-1326 164 35 𝛼 𝛼 PROPN cana-1326 164 36 + + X cana-1326 164 37 𝛼2 𝛼2 ADJ cana-1326 164 38 + + NOUN cana-1326 164 39 β‹―+ β‹―+ NOUN cana-1326 164 40 π›Όπ‘šβˆ’1) π›Όπ‘šβˆ’1) X cana-1326 164 41 οΏ½ οΏ½ PROPN cana-1326 164 42 d(π‘₯0 d(π‘₯0 PROPN cana-1326 164 43 , , PUNCT cana-1326 164 44 π‘₯1 π‘₯1 NOUN cana-1326 164 45 ) ) PUNCT cana-1326 164 46 ∴ ∴ PROPN cana-1326 164 47 β€–d(π‘₯𝑛 β€–d(π‘₯𝑛 PROPN cana-1326 164 48 , , PUNCT cana-1326 164 49 π‘₯π‘š+𝑛)β€– π‘₯π‘š+𝑛)β€– ADJ cana-1326 164 50 ≀ ≀ NUM cana-1326 164 51 𝛼𝑛 𝛼𝑛 ADP cana-1326 164 52 1βˆ’π›Όπ‘š 1βˆ’π›Όπ‘š NUM cana-1326 164 53 1βˆ’π›Ό 1βˆ’π›Ό NUM cana-1326 164 54 οΏ½ οΏ½ PROPN cana-1326 164 55 d(π‘₯0 d(π‘₯0 PROPN cana-1326 164 56 , , PUNCT cana-1326 164 57 π‘₯1)β†’0 π‘₯1)β†’0 VERB cana-1326 164 58 as as ADP cana-1326 164 59 m m PROPN cana-1326 164 60 , , PUNCT cana-1326 164 61 nβ†’βˆž nβ†’βˆž X cana-1326 164 62 ∴ ∴ PROPN cana-1326 164 63 * * PUNCT cana-1326 164 64 π‘₯𝑛+ π‘₯𝑛+ PROPN cana-1326 164 65 οΏ½ οΏ½ PROPN cana-1326 164 66 is be AUX cana-1326 164 67 οΏ½ οΏ½ PROPN cana-1326 164 68 a a DET cana-1326 164 69 οΏ½ οΏ½ PROPN cana-1326 164 70 cauchy cauchy NOUN cana-1326 164 71 οΏ½ οΏ½ PROPN cana-1326 164 72 sequence sequence NOUN cana-1326 164 73 οΏ½ οΏ½ PROPN cana-1326 164 74 in in ADP cana-1326 164 75 οΏ½ οΏ½ PROPN cana-1326 164 76 x. x. PROPN cana-1326 164 77 since since SCONJ cana-1326 164 78 οΏ½ οΏ½ PROPN cana-1326 164 79 x x SYM cana-1326 164 80 οΏ½ οΏ½ PROPN cana-1326 164 81 is be AUX cana-1326 164 82 οΏ½ οΏ½ NOUN cana-1326 164 83 complete complete ADJ cana-1326 164 84 οΏ½ οΏ½ PROPN cana-1326 164 85 there there PRON cana-1326 164 86 οΏ½ οΏ½ NOUN cana-1326 164 87 exist exist VERB cana-1326 164 88 οΏ½ οΏ½ NOUN cana-1326 164 89 x x SYM cana-1326 164 90 ∈ ∈ PROPN cana-1326 164 91 x x SYM cana-1326 164 92 οΏ½ οΏ½ PROPN cana-1326 164 93 such such ADJ cana-1326 164 94 οΏ½ οΏ½ PROPN cana-1326 164 95 that that PRON cana-1326 164 96 οΏ½ οΏ½ VERB cana-1326 164 97 π‘₯𝑛 π‘₯𝑛 PROPN cana-1326 164 98 β†’ β†’ SYM cana-1326 164 99 x x SYM cana-1326 164 100 οΏ½ οΏ½ PROPN cana-1326 164 101 as as ADP cana-1326 164 102 οΏ½ οΏ½ NOUN cana-1326 164 103 n n CCONJ cana-1326 164 104 β†’ β†’ SYM cana-1326 164 105 ∞. ∞. PROPN cana-1326 164 106 thus thus ADV cana-1326 164 107 limπ‘›β†’βˆž limπ‘›β†’βˆž NOUN cana-1326 164 108 𝑆(π‘₯2𝑛 𝑆(π‘₯2𝑛 ADV cana-1326 164 109 ) ) PUNCT cana-1326 165 1 = = SYM cana-1326 165 2 limπ‘›β†’βˆž limπ‘›β†’βˆž PROPN cana-1326 165 3 𝑇(π‘₯2𝑛+1 𝑇(π‘₯2𝑛+1 PROPN cana-1326 165 4 ) ) PUNCT cana-1326 166 1 = = PUNCT cana-1326 166 2 π‘₯. π‘₯. VERB cana-1326 166 3 thus thus ADV cana-1326 166 4 from from ADP cana-1326 166 5 ( ( PUNCT cana-1326 166 6 2.1 2.1 NUM cana-1326 166 7 ) ) PUNCT cana-1326 166 8 , , PUNCT cana-1326 166 9 we we PRON cana-1326 166 10 have have VERB cana-1326 166 11 d(sx d(sx NOUN cana-1326 166 12 , , PUNCT cana-1326 166 13 x x X cana-1326 166 14 ) ) PUNCT cana-1326 166 15 ≲𝑖2d(sx ≲𝑖2d(sx NOUN cana-1326 166 16 , , PUNCT cana-1326 166 17 tπ‘₯2π‘˜+1)+d(tπ‘₯2π‘˜+1,x tπ‘₯2π‘˜+1)+d(tπ‘₯2π‘˜+1,x NOUN cana-1326 166 18 ) ) PUNCT cana-1326 166 19 ≲𝑖2 ≲𝑖2 NOUN cana-1326 166 20 οΏ½ οΏ½ NOUN cana-1326 166 21 𝛼max[d(x 𝛼max[d(x NUM cana-1326 166 22 , , PUNCT cana-1326 166 23 π‘₯2π‘˜+1 π‘₯2π‘˜+1 PROPN cana-1326 166 24 ) ) PUNCT cana-1326 166 25 , , PUNCT cana-1326 166 26 𝑑(π‘₯,𝑆(π‘₯))𝑑(π‘₯2π‘˜+1,𝑇(π‘₯2π‘˜+1 𝑑(π‘₯,𝑆(π‘₯))𝑑(π‘₯2π‘˜+1,𝑇(π‘₯2π‘˜+1 NOUN cana-1326 166 27 ) ) PUNCT cana-1326 166 28 ) ) PUNCT cana-1326 167 1 1+d(sx,𝑇π‘₯2π‘˜+1 1+d(sx,𝑇π‘₯2π‘˜+1 X cana-1326 167 2 ) ) PUNCT cana-1326 167 3 οΏ½ οΏ½ PROPN cana-1326 167 4 ] ] PUNCT cana-1326 167 5 ) ) PUNCT cana-1326 168 1 + + ADJ cana-1326 168 2 d(π‘₯2π‘˜+2,x d(π‘₯2π‘˜+2,x NOUN cana-1326 168 3 ) ) PUNCT cana-1326 168 4 ≲𝑖2 ≲𝑖2 NOUN cana-1326 168 5 οΏ½ οΏ½ NOUN cana-1326 168 6 𝛼d(x 𝛼d(x NOUN cana-1326 168 7 , , PUNCT cana-1326 168 8 π‘₯2π‘˜+1 π‘₯2π‘˜+1 X cana-1326 168 9 ) ) PUNCT cana-1326 169 1 + + NOUN cana-1326 169 2 d(π‘₯2π‘˜+2,x d(π‘₯2π‘˜+2,x NOUN cana-1326 169 3 ) ) PUNCT cana-1326 169 4 ∴ ∴ PROPN cana-1326 169 5 β€–d(sx β€–d(sx PROPN cana-1326 169 6 , , PUNCT cana-1326 169 7 x) x) PROPN cana-1326 169 8 οΏ½ οΏ½ PROPN cana-1326 169 9 β€– β€– PROPN cana-1326 169 10 ≀ ≀ PROPN cana-1326 169 11 οΏ½ οΏ½ PROPN cana-1326 169 12 𝛼‖d(x 𝛼‖d(x PROPN cana-1326 169 13 , , PUNCT cana-1326 169 14 π‘₯2π‘˜+1)β€– π‘₯2π‘˜+1)β€– NOUN cana-1326 169 15 + + CCONJ cana-1326 169 16 β€–d(π‘₯2π‘˜+2 β€–d(π‘₯2π‘˜+2 PROPN cana-1326 169 17 , , PUNCT cana-1326 169 18 x) x) PROPN cana-1326 169 19 οΏ½ οΏ½ PROPN cana-1326 169 20 οΏ½ οΏ½ PROPN cana-1326 169 21 β€– β€– PROPN cana-1326 169 22 β†’0 β†’0 PUNCT cana-1326 169 23 as as ADP cana-1326 169 24 nβ†’βˆž. nβ†’βˆž. ADJ cana-1326 169 25 thus thus ADV cana-1326 169 26 β€–d(sx β€–d(sx PROPN cana-1326 169 27 , , PUNCT cana-1326 169 28 x)β€– x)β€– NOUN cana-1326 169 29 = = NOUN cana-1326 169 30 0 0 X cana-1326 169 31 . . PUNCT cana-1326 170 1 so so ADV cana-1326 170 2 οΏ½ οΏ½ NOUN cana-1326 170 3 𝑆(π‘₯ 𝑆(π‘₯ NUM cana-1326 170 4 ) ) PUNCT cana-1326 170 5 = = PUNCT cana-1326 170 6 π‘₯. π‘₯. ADV cana-1326 170 7 similarly similarly ADV cana-1326 170 8 , , PUNCT cana-1326 170 9 we we PRON cana-1326 170 10 can can AUX cana-1326 170 11 prove prove VERB cana-1326 170 12 t(x)=x t(x)=x PROPN cana-1326 170 13 . . PUNCT cana-1326 171 1 thus thus ADV cana-1326 171 2 x x PRON cana-1326 171 3 is be AUX cana-1326 171 4 a a DET cana-1326 171 5 common common ADJ cana-1326 171 6 fixed fix VERB cana-1326 171 7 point point NOUN cana-1326 171 8 of of ADP cana-1326 171 9 s s PRON cana-1326 171 10 and and CCONJ cana-1326 171 11 t. t. NOUN cana-1326 171 12 now now ADV cana-1326 171 13 for for ADP cana-1326 171 14 uniqueness uniqueness NOUN cana-1326 171 15 let let VERB cana-1326 171 16 us we PRON cana-1326 171 17 assume assume VERB cana-1326 171 18 that that SCONJ cana-1326 171 19 π‘₯βˆ— π‘₯βˆ— PROPN cana-1326 171 20 ∈ ∈ PROPN cana-1326 171 21 𝑋 𝑋 PROPN cana-1326 171 22 is be AUX cana-1326 171 23 another another DET cana-1326 171 24 fixed fix VERB cana-1326 171 25 point point NOUN cana-1326 171 26 of of ADP cana-1326 171 27 s s PRON cana-1326 171 28 and and CCONJ cana-1326 171 29 t. t. PROPN cana-1326 171 30 then then ADV cana-1326 171 31 d(x d(x PROPN cana-1326 171 32 , , PUNCT cana-1326 171 33 π‘₯βˆ— π‘₯βˆ— PROPN cana-1326 171 34 ) ) PUNCT cana-1326 172 1 = = SYM cana-1326 172 2 𝑑(𝑆π‘₯ 𝑑(𝑆π‘₯ ADJ cana-1326 172 3 , , PUNCT cana-1326 172 4 𝑇π‘₯ 𝑇π‘₯ NOUN cana-1326 172 5 ) ) PUNCT cana-1326 172 6 οΏ½ οΏ½ NOUN cana-1326 172 7 ≲𝑖2 ≲𝑖2 NOUN cana-1326 172 8 𝛼max[d(x 𝛼max[d(x NOUN cana-1326 172 9 , , PUNCT cana-1326 172 10 π‘₯βˆ— π‘₯βˆ— PROPN cana-1326 172 11 ) ) PUNCT cana-1326 172 12 , , PUNCT cana-1326 172 13 𝑑(π‘₯,𝑆(π‘₯))𝑑(π‘₯βˆ—,𝑇(π‘₯βˆ— 𝑑(π‘₯,𝑆(π‘₯))𝑑(π‘₯βˆ—,𝑇(π‘₯βˆ— PROPN cana-1326 172 14 ) ) PUNCT cana-1326 172 15 ) ) PUNCT cana-1326 173 1 1+d(sx,𝑇π‘₯βˆ— 1+d(sx,𝑇π‘₯βˆ— X cana-1326 173 2 ) ) PUNCT cana-1326 173 3 οΏ½ οΏ½ PROPN cana-1326 173 4 ] ] PUNCT cana-1326 173 5 ≲𝑖2 ≲𝑖2 NOUN cana-1326 173 6 𝛼d(x 𝛼d(x NOUN cana-1326 173 7 , , PUNCT cana-1326 173 8 π‘₯βˆ— π‘₯βˆ— PROPN cana-1326 173 9 ) ) PUNCT cana-1326 173 10 β‡’(1-𝛼 β‡’(1-𝛼 PUNCT cana-1326 173 11 ) ) PUNCT cana-1326 173 12 d(x d(x PROPN cana-1326 173 13 , , PUNCT cana-1326 173 14 π‘₯βˆ— π‘₯βˆ— PROPN cana-1326 173 15 ) ) PUNCT cana-1326 173 16 οΏ½ οΏ½ PROPN cana-1326 174 1 ≲𝑖2 ≲𝑖2 NOUN cana-1326 174 2 0 0 NUM cana-1326 174 3 β‡’(1-𝛼)β€–d(x β‡’(1-𝛼)β€–d(x NOUN cana-1326 174 4 , , PUNCT cana-1326 174 5 π‘₯βˆ—) π‘₯βˆ—) PROPN cana-1326 174 6 οΏ½ οΏ½ PROPN cana-1326 174 7 β€– β€– ADJ cana-1326 174 8 ≀ ≀ NOUN cana-1326 174 9 0 0 NUM cana-1326 174 10 β‡’ β‡’ NOUN cana-1326 174 11 d(x d(x PROPN cana-1326 174 12 , , PUNCT cana-1326 174 13 π‘₯βˆ— π‘₯βˆ— PROPN cana-1326 174 14 ) ) PUNCT cana-1326 175 1 = = SYM cana-1326 175 2 0 0 NUM cana-1326 175 3 β‡’ β‡’ NOUN cana-1326 175 4 π‘₯ π‘₯ X cana-1326 175 5 = = PUNCT cana-1326 175 6 π‘₯βˆ—. π‘₯βˆ—. NOUN cana-1326 175 7 communications communication NOUN cana-1326 175 8 on on ADP cana-1326 175 9 applied apply VERB cana-1326 175 10 nonlinear nonlinear ADJ cana-1326 175 11 analysis analysis NOUN cana-1326 175 12 issn issn NOUN cana-1326 175 13 : : PUNCT cana-1326 175 14 1074 1074 NUM cana-1326 175 15 - - PUNCT cana-1326 175 16 133x 133x NUM cana-1326 175 17 vol vol NOUN cana-1326 175 18 31 31 NUM cana-1326 175 19 no no NOUN cana-1326 175 20 . . PUNCT cana-1326 176 1 7s 7 NOUN cana-1326 176 2 ( ( PUNCT cana-1326 176 3 2024 2024 NUM cana-1326 176 4 ) ) PUNCT cana-1326 176 5 474 474 NUM cana-1326 176 6 https://internationalpubls.com https://internationalpubls.com X cana-1326 176 7 this this PRON cana-1326 176 8 completes complete VERB cana-1326 176 9 the the DET cana-1326 176 10 proof proof NOUN cana-1326 176 11 of of ADP cana-1326 176 12 the the DET cana-1326 176 13 theorem theorem PROPN cana-1326 176 14 . . PUNCT cana-1326 176 15 theorem theorem NOUN cana-1326 176 16 3 3 NUM cana-1326 176 17 : : PUNCT cana-1326 176 18 let let VERB cana-1326 176 19 ( ( PUNCT cana-1326 176 20 𝑋 𝑋 PROPN cana-1326 176 21 οΏ½ οΏ½ PROPN cana-1326 176 22 , , PUNCT cana-1326 176 23 𝑑 𝑑 NOUN cana-1326 176 24 οΏ½ οΏ½ NOUN cana-1326 176 25 ) ) PUNCT cana-1326 176 26 be be VERB cana-1326 176 27 a a DET cana-1326 176 28 complete complete ADJ cana-1326 176 29 bicomplex bicomplex NOUN cana-1326 176 30 valued value VERB cana-1326 176 31 𝑏 𝑏 DET cana-1326 176 32 οΏ½ οΏ½ NOUN cana-1326 176 33 -metric -metric ADJ cana-1326 176 34 space space NOUN cana-1326 176 35 with with ADP cana-1326 176 36 the the DET cana-1326 176 37 coefficient coefficient NOUN cana-1326 176 38 𝑠 𝑠 PROPN cana-1326 176 39 οΏ½ οΏ½ PROPN cana-1326 176 40 β‰₯1 β‰₯1 PROPN cana-1326 176 41 and and CCONJ cana-1326 176 42 let let VERB cana-1326 176 43 𝑆 𝑆 PROPN cana-1326 176 44 οΏ½ οΏ½ PROPN cana-1326 176 45 , , PUNCT cana-1326 176 46 t t PROPN cana-1326 176 47 : : PUNCT cana-1326 176 48 𝑋 𝑋 PROPN cana-1326 176 49 οΏ½ οΏ½ PROPN cana-1326 176 50 β†’ β†’ SYM cana-1326 176 51 𝑋 𝑋 PROPN cana-1326 176 52 οΏ½ οΏ½ PROPN cana-1326 176 53 be be AUX cana-1326 176 54 mappings mapping NOUN cana-1326 176 55 satisfying satisfy VERB cana-1326 176 56 d(sx d(sx NOUN cana-1326 176 57 , , PUNCT cana-1326 176 58 ty ty NUM cana-1326 176 59 ) ) PUNCT cana-1326 176 60 ≲𝑖2ad(x ≲𝑖2ad(x ADV cana-1326 176 61 , , PUNCT cana-1326 176 62 y)+b y)+b PROPN cana-1326 176 63 𝑑(π‘₯,𝑆π‘₯)𝑑(𝑦,𝑇𝑦 𝑑(π‘₯,𝑆π‘₯)𝑑(𝑦,𝑇𝑦 NUM cana-1326 176 64 ) ) PUNCT cana-1326 177 1 d d X cana-1326 177 2 οΏ½ οΏ½ PROPN cana-1326 177 3 (x (x PROPN cana-1326 177 4 , , PUNCT cana-1326 177 5 ty) ty) PROPN cana-1326 177 6 οΏ½ οΏ½ PROPN cana-1326 177 7 + + NOUN cana-1326 177 8 οΏ½ οΏ½ PROPN cana-1326 177 9 d d PROPN cana-1326 177 10 οΏ½ οΏ½ PROPN cana-1326 177 11 (y (y NOUN cana-1326 177 12 , , PUNCT cana-1326 177 13 sx) sx) X cana-1326 177 14 οΏ½ οΏ½ PROPN cana-1326 177 15 + + NOUN cana-1326 177 16 οΏ½ οΏ½ PROPN cana-1326 177 17 d d NOUN cana-1326 177 18 οΏ½ οΏ½ PROPN cana-1326 177 19 (x (x PROPN cana-1326 177 20 , , PUNCT cana-1326 177 21 y y PROPN cana-1326 177 22 ) ) PUNCT cana-1326 177 23 … … PUNCT cana-1326 177 24 … … PUNCT cana-1326 177 25 … … PUNCT cana-1326 177 26 ( ( PUNCT cana-1326 177 27 3.1 3.1 NUM cana-1326 177 28 ) ) PUNCT cana-1326 177 29 for for ADP cana-1326 177 30 all all DET cana-1326 177 31 π‘₯ π‘₯ PROPN cana-1326 177 32 οΏ½ οΏ½ PROPN cana-1326 177 33 , , PUNCT cana-1326 177 34 𝑦 𝑦 NOUN cana-1326 177 35 οΏ½ οΏ½ PROPN cana-1326 177 36 ∈ ∈ PROPN cana-1326 177 37 𝑋 𝑋 PROPN cana-1326 177 38 οΏ½ οΏ½ PROPN cana-1326 177 39 , , PUNCT cana-1326 177 40 such such ADJ cana-1326 177 41 that that SCONJ cana-1326 177 42 xβ‰  xβ‰  PROPN cana-1326 177 43 𝑦 𝑦 PROPN cana-1326 177 44 , , PUNCT cana-1326 177 45 𝑑 𝑑 PROPN cana-1326 177 46 οΏ½ οΏ½ PROPN cana-1326 177 47 (π‘₯ (π‘₯ PROPN cana-1326 177 48 οΏ½ οΏ½ PROPN cana-1326 177 49 , , PUNCT cana-1326 177 50 𝑇 𝑇 PROPN cana-1326 177 51 οΏ½ οΏ½ PROPN cana-1326 177 52 𝑦 𝑦 NOUN cana-1326 177 53 οΏ½ οΏ½ PROPN cana-1326 177 54 )+𝑑 )+𝑑 PROPN cana-1326 177 55 οΏ½ οΏ½ PROPN cana-1326 177 56 (𝑦 (𝑦 PROPN cana-1326 177 57 οΏ½ οΏ½ PROPN cana-1326 177 58 , , PUNCT cana-1326 177 59 𝑆 𝑆 PROPN cana-1326 177 60 οΏ½ οΏ½ PROPN cana-1326 177 61 π‘₯ π‘₯ PROPN cana-1326 177 62 οΏ½ οΏ½ PROPN cana-1326 177 63 )+𝑑 )+𝑑 PROPN cana-1326 177 64 οΏ½ οΏ½ PROPN cana-1326 177 65 (π‘₯ (π‘₯ PROPN cana-1326 177 66 οΏ½ οΏ½ PROPN cana-1326 177 67 , , PUNCT cana-1326 177 68 𝑦 𝑦 NOUN cana-1326 177 69 οΏ½ οΏ½ NOUN cana-1326 177 70 ) ) PUNCT cana-1326 177 71 β‰  β‰  PROPN cana-1326 177 72 0 0 NUM cana-1326 177 73 , , PUNCT cana-1326 177 74 where where SCONJ cana-1326 177 75 a a DET cana-1326 177 76 , , PUNCT cana-1326 177 77 b b NOUN cana-1326 177 78 are be AUX cana-1326 177 79 nonnegative nonnegative ADJ cana-1326 177 80 reals real NOUN cana-1326 177 81 with with ADP cana-1326 177 82 a+ a+ DET cana-1326 177 83 √2𝑠 √2𝑠 NUM cana-1326 177 84 οΏ½ οΏ½ NOUN cana-1326 177 85 b b NOUN cana-1326 177 86 < < X cana-1326 177 87 1 1 NUM cana-1326 177 88 or or CCONJ cana-1326 177 89 𝑑 𝑑 PROPN cana-1326 177 90 οΏ½ οΏ½ PROPN cana-1326 177 91 (𝑆 (𝑆 SYM cana-1326 177 92 οΏ½ οΏ½ PROPN cana-1326 177 93 π‘₯ π‘₯ NOUN cana-1326 177 94 οΏ½ οΏ½ PROPN cana-1326 177 95 , , PUNCT cana-1326 177 96 𝑇 𝑇 PROPN cana-1326 177 97 οΏ½ οΏ½ PROPN cana-1326 177 98 𝑦 𝑦 NOUN cana-1326 177 99 οΏ½ οΏ½ PROPN cana-1326 177 100 ) ) PUNCT cana-1326 177 101 = = SYM cana-1326 177 102 0 0 PUNCT cana-1326 177 103 if if SCONJ cana-1326 177 104 𝑑 𝑑 PROPN cana-1326 177 105 οΏ½ οΏ½ PROPN cana-1326 177 106 (π‘₯ (π‘₯ PROPN cana-1326 177 107 οΏ½ οΏ½ PROPN cana-1326 177 108 , , PUNCT cana-1326 177 109 𝑇 𝑇 PROPN cana-1326 177 110 οΏ½ οΏ½ PROPN cana-1326 177 111 𝑦 𝑦 NOUN cana-1326 177 112 οΏ½ οΏ½ PROPN cana-1326 177 113 ) ) PUNCT cana-1326 177 114 + + CCONJ cana-1326 177 115 𝑑 𝑑 PROPN cana-1326 177 116 οΏ½ οΏ½ PROPN cana-1326 177 117 (𝑦 (𝑦 NOUN cana-1326 177 118 οΏ½ οΏ½ PROPN cana-1326 177 119 , , PUNCT cana-1326 177 120 𝑆 𝑆 PROPN cana-1326 177 121 οΏ½ οΏ½ PROPN cana-1326 177 122 π‘₯ π‘₯ NOUN cana-1326 177 123 οΏ½ οΏ½ PROPN cana-1326 177 124 ) ) PUNCT cana-1326 177 125 + + CCONJ cana-1326 177 126 𝑑 𝑑 PROPN cana-1326 177 127 οΏ½ οΏ½ PROPN cana-1326 177 128 (π‘₯ (π‘₯ PROPN cana-1326 177 129 οΏ½ οΏ½ PROPN cana-1326 177 130 , , PUNCT cana-1326 177 131 𝑦 𝑦 NOUN cana-1326 177 132 οΏ½ οΏ½ NOUN cana-1326 177 133 ) ) PUNCT cana-1326 177 134 = = SYM cana-1326 178 1 0 0 X cana-1326 178 2 . . PUNCT cana-1326 179 1 then then ADV cana-1326 179 2 𝑆 𝑆 PROPN cana-1326 179 3 οΏ½ οΏ½ PROPN cana-1326 179 4 and and CCONJ cana-1326 179 5 𝑇 𝑇 PROPN cana-1326 179 6 οΏ½ οΏ½ PROPN cana-1326 179 7 have have VERB cana-1326 179 8 a a DET cana-1326 179 9 unique unique ADJ cana-1326 179 10 common common ADJ cana-1326 179 11 fixed fix VERB cana-1326 179 12 point point NOUN cana-1326 179 13 . . PUNCT cana-1326 180 1 proof proof NOUN cana-1326 180 2 : : PUNCT cana-1326 180 3 let let VERB cana-1326 180 4 π‘₯0 π‘₯0 PROPN cana-1326 180 5 οΏ½ οΏ½ PROPN cana-1326 180 6 be be AUX cana-1326 180 7 an an DET cana-1326 180 8 arbitrary arbitrary ADJ cana-1326 180 9 point point NOUN cana-1326 180 10 in in ADP cana-1326 180 11 x x PUNCT cana-1326 180 12 and and CCONJ cana-1326 180 13 define define VERB cana-1326 180 14 a a DET cana-1326 180 15 sequence sequence NOUN cana-1326 180 16 { { PUNCT cana-1326 180 17 π‘₯𝑛 π‘₯𝑛 NOUN cana-1326 180 18 } } PUNCT cana-1326 180 19 in in ADP cana-1326 180 20 x x INTJ cana-1326 180 21 such such ADJ cana-1326 180 22 that that PRON cana-1326 180 23 π‘₯2𝑛+1 π‘₯2𝑛+1 ADJ cana-1326 180 24 = = SYM cana-1326 180 25 𝑆π‘₯2𝑛 𝑆π‘₯2𝑛 PROPN cana-1326 180 26 and and CCONJ cana-1326 180 27 π‘₯2𝑛+2 π‘₯2𝑛+2 NOUN cana-1326 180 28 = = NOUN cana-1326 180 29 𝑇π‘₯2𝑛+1 𝑇π‘₯2𝑛+1 NOUN cana-1326 180 30 where where SCONJ cana-1326 180 31 n=0,1,2 n=0,1,2 NUM cana-1326 180 32 … … NUM cana-1326 180 33 . . PUNCT cana-1326 180 34 ... ... PUNCT cana-1326 181 1 … … PUNCT cana-1326 181 2 ( ( PUNCT cana-1326 181 3 3.2 3.2 NUM cana-1326 181 4 ) ) PUNCT cana-1326 181 5 . . PUNCT cana-1326 182 1 now now ADV cana-1326 182 2 , , PUNCT cana-1326 182 3 we we PRON cana-1326 182 4 show show VERB cana-1326 182 5 that that SCONJ cana-1326 182 6 the the DET cana-1326 182 7 sequence sequence NOUN cana-1326 182 8 { { PUNCT cana-1326 182 9 π‘₯𝑛+ π‘₯𝑛+ PROPN cana-1326 182 10 is be AUX cana-1326 182 11 cauchy cauchy PROPN cana-1326 182 12 . . PUNCT cana-1326 183 1 let let VERB cana-1326 183 2 x= x= PROPN cana-1326 183 3 οΏ½ οΏ½ PROPN cana-1326 183 4 x2n x2n PROPN cana-1326 183 5 and and CCONJ cana-1326 183 6 y y PROPN cana-1326 183 7 = = PROPN cana-1326 183 8 x2n+1 x2n+1 PROPN cana-1326 183 9 in in ADP cana-1326 183 10 ( ( PUNCT cana-1326 183 11 3.2 3.2 NUM cana-1326 183 12 ) ) PUNCT cana-1326 183 13 ; ; PUNCT cana-1326 183 14 we we PRON cana-1326 183 15 have have VERB cana-1326 183 16 d(x2n+1 d(x2n+1 NOUN cana-1326 183 17 , , PUNCT cana-1326 183 18 x2n+2 x2n+2 PUNCT cana-1326 183 19 ) ) PUNCT cana-1326 184 1 = = SYM cana-1326 184 2 d(sx2n d(sx2n NOUN cana-1326 184 3 , , PUNCT cana-1326 184 4 tx2n+1 tx2n+1 NOUN cana-1326 184 5 ) ) PUNCT cana-1326 184 6 ≲𝑖2 ≲𝑖2 PROPN cana-1326 184 7 𝐴 𝐴 PROPN cana-1326 184 8 οΏ½ οΏ½ PROPN cana-1326 184 9 d(x2n d(x2n PROPN cana-1326 184 10 , , PUNCT cana-1326 184 11 x2n+1 x2n+1 PUNCT cana-1326 184 12 ) ) PUNCT cana-1326 185 1 + + CCONJ cana-1326 185 2 𝐡 𝐡 NOUN cana-1326 185 3 d(x2n+1,tx2n+1)d(x2n d(x2n+1,tx2n+1)d(x2n NOUN cana-1326 185 4 , , PUNCT cana-1326 185 5 sx2n sx2n PROPN cana-1326 185 6 ) ) PUNCT cana-1326 185 7 d(x2n d(x2n PROPN cana-1326 185 8 , , PUNCT cana-1326 185 9 tx2n+1)+𝑑(x2n+1,sx2n)+d(x2n tx2n+1)+𝑑(x2n+1,sx2n)+d(x2n PROPN cana-1326 185 10 , , PUNCT cana-1326 185 11 x2n+1 x2n+1 PROPN cana-1326 185 12 ) ) PUNCT cana-1326 185 13 ( ( PUNCT cana-1326 185 14 3.3 3.3 NUM cana-1326 185 15 ) ) PUNCT cana-1326 185 16 ≲𝑖2 ≲𝑖2 PROPN cana-1326 185 17 𝐴 𝐴 PROPN cana-1326 185 18 οΏ½ οΏ½ PROPN cana-1326 185 19 d(x2n d(x2n PROPN cana-1326 185 20 , , PUNCT cana-1326 185 21 x2n+1 x2n+1 PUNCT cana-1326 185 22 ) ) PUNCT cana-1326 186 1 + + CCONJ cana-1326 186 2 𝐡 𝐡 NOUN cana-1326 186 3 d(x2n+1,x2n+2)d(x2n d(x2n+1,x2n+2)d(x2n PROPN cana-1326 186 4 , , PUNCT cana-1326 186 5 x2n+1 x2n+1 PROPN cana-1326 186 6 ) ) PUNCT cana-1326 186 7 d(x2n d(x2n PROPN cana-1326 186 8 , , PUNCT cana-1326 186 9 x2n+2)+𝑑(x2n+1,x2n+1)+d(x2n x2n+2)+𝑑(x2n+1,x2n+1)+d(x2n PROPN cana-1326 186 10 , , PUNCT cana-1326 186 11 x2n+1 x2n+1 PROPN cana-1326 186 12 ) ) PUNCT cana-1326 186 13 this this PRON cana-1326 186 14 implies imply VERB cana-1326 186 15 that that PRON cana-1326 186 16 β€–d(x2n+1 β€–d(x2n+1 PROPN cana-1326 186 17 , , PUNCT cana-1326 186 18 x2n+2)β€– x2n+2)β€– PUNCT cana-1326 186 19 ≀ ≀ NUM cana-1326 186 20 𝐴‖d(x2n 𝐴‖d(x2n PROPN cana-1326 186 21 , , PUNCT cana-1326 186 22 x2n+1)β€– x2n+1)β€– PROPN cana-1326 187 1 + + NOUN cana-1326 187 2 √2𝐡 √2𝐡 PROPN cana-1326 187 3 β€–d(x2n+1,x2n+2)β€–β€–d(x2n β€–d(x2n+1,x2n+2)β€–β€–d(x2n NOUN cana-1326 187 4 , , PUNCT cana-1326 187 5 x2n+1)β€– x2n+1)β€– PROPN cana-1326 187 6 β€–d(x2n β€–d(x2n PROPN cana-1326 187 7 , , PUNCT cana-1326 187 8 x2n+2)β€–+β€–d(x2n x2n+2)β€–+β€–d(x2n PROPN cana-1326 187 9 , , PUNCT cana-1326 187 10 x2n+1)β€– x2n+1)β€– PROPN cana-1326 187 11 ( ( PUNCT cana-1326 187 12 3.4 3.4 NUM cana-1326 187 13 ) ) PUNCT cana-1326 187 14 as as ADP cana-1326 187 15 β€–d(x2n+1 β€–d(x2n+1 PROPN cana-1326 187 16 , , PUNCT cana-1326 187 17 x2n+2)β€– x2n+2)β€– X cana-1326 187 18 ≀ ≀ PROPN cana-1326 187 19 𝑠(β€–d(x2n+1 𝑠(β€–d(x2n+1 PROPN cana-1326 187 20 , , PUNCT cana-1326 187 21 x2n)β€– x2n)β€– X cana-1326 187 22 + + CCONJ cana-1326 187 23 β€–d(x2n β€–d(x2n PROPN cana-1326 187 24 , , PUNCT cana-1326 187 25 x2n+2)β€– x2n+2)β€– NOUN cana-1326 187 26 ) ) PUNCT cana-1326 187 27 ( ( PUNCT cana-1326 187 28 3.5 3.5 NUM cana-1326 187 29 ) ) PUNCT cana-1326 187 30 therefore therefore ADV cana-1326 187 31 β€–d(x2n+1 β€–d(x2n+1 PROPN cana-1326 187 32 , , PUNCT cana-1326 187 33 x2n+2)β€– x2n+2)β€– PUNCT cana-1326 187 34 ≀ ≀ NUM cana-1326 187 35 𝐴‖d(x2n 𝐴‖d(x2n PROPN cana-1326 187 36 , , PUNCT cana-1326 187 37 x2n+1)β€– x2n+1)β€– PROPN cana-1326 187 38 + + PROPN cana-1326 188 1 √2𝑠𝐡‖d(x2n √2𝑠𝐡‖d(x2n PROPN cana-1326 188 2 , , PUNCT cana-1326 188 3 x2n+1)β€– x2n+1)β€– PROPN cana-1326 188 4 ≀ ≀ PROPN cana-1326 188 5 ( ( PUNCT cana-1326 188 6 𝐴 𝐴 PROPN cana-1326 188 7 + + CCONJ cana-1326 188 8 √2𝑠𝐡)β€–d(x2n √2𝑠𝐡)β€–d(x2n NOUN cana-1326 188 9 , , PUNCT cana-1326 188 10 x2n+1)β€– x2n+1)β€– PROPN cana-1326 188 11 ( ( PUNCT cana-1326 188 12 3.6 3.6 NUM cana-1326 188 13 ) ) PUNCT cana-1326 188 14 similarly similarly ADV cana-1326 188 15 we we PRON cana-1326 188 16 get get VERB cana-1326 188 17 , , PUNCT cana-1326 188 18 β€–d(x2n+2 β€–d(x2n+2 PROPN cana-1326 188 19 , , PUNCT cana-1326 188 20 x2n+3)β€– x2n+3)β€– PROPN cana-1326 188 21 ≀ ≀ PROPN cana-1326 188 22 ( ( PUNCT cana-1326 188 23 𝐴 𝐴 PROPN cana-1326 188 24 + + CCONJ cana-1326 188 25 √2𝑠𝐡) √2𝑠𝐡) PROPN cana-1326 188 26 οΏ½ οΏ½ PROPN cana-1326 188 27 β€–d(x2n+1 β€–d(x2n+1 PROPN cana-1326 188 28 , , PUNCT cana-1326 188 29 x2n+2)β€– x2n+2)β€– PROPN cana-1326 188 30 ( ( PUNCT cana-1326 188 31 3.7 3.7 NUM cana-1326 188 32 ) ) PUNCT cana-1326 188 33 since since SCONJ cana-1326 188 34 ( ( PUNCT cana-1326 188 35 𝐴 𝐴 PROPN cana-1326 188 36 + + CCONJ cana-1326 188 37 √2𝑠𝐡 √2𝑠𝐡 NOUN cana-1326 188 38 ) ) PUNCT cana-1326 188 39 < < X cana-1326 189 1 1 1 X cana-1326 189 2 . . X cana-1326 189 3 οΏ½ οΏ½ PROPN cana-1326 189 4 therefore therefore ADV cana-1326 189 5 with with ADP cana-1326 189 6 ( ( PUNCT cana-1326 189 7 𝐴 𝐴 PROPN cana-1326 189 8 + + CCONJ cana-1326 189 9 √2𝑠𝐡 √2𝑠𝐡 NOUN cana-1326 189 10 ) ) PUNCT cana-1326 189 11 = = PUNCT cana-1326 190 1 πœ† πœ† PUNCT cana-1326 190 2 < < X cana-1326 190 3 1 1 NUM cana-1326 190 4 , , PUNCT cana-1326 190 5 and and CCONJ cana-1326 190 6 for for ADP cana-1326 190 7 all all DET cana-1326 190 8 nβ‰₯ nβ‰₯ NOUN cana-1326 190 9 0 0 NUM cana-1326 190 10 , , PUNCT cana-1326 190 11 and and CCONJ cana-1326 190 12 consequently consequently ADV cana-1326 190 13 , , PUNCT cana-1326 190 14 we we PRON cana-1326 190 15 have have VERB cana-1326 190 16 β€–d(x2n+1 β€–d(x2n+1 PROPN cana-1326 190 17 , , PUNCT cana-1326 190 18 x2n+2)β€– x2n+2)β€– X cana-1326 190 19 ≀ ≀ NUM cana-1326 190 20 οΏ½ οΏ½ PROPN cana-1326 190 21 πœ† πœ† NOUN cana-1326 190 22 οΏ½ οΏ½ NOUN cana-1326 190 23 β€–d(x2n β€–d(x2n NOUN cana-1326 190 24 , , PUNCT cana-1326 190 25 x2n+1)β€– x2n+1)β€– PROPN cana-1326 190 26 ≀ ≀ PROPN cana-1326 190 27 οΏ½ οΏ½ PROPN cana-1326 190 28 πœ† πœ† NOUN cana-1326 190 29 οΏ½ οΏ½ PROPN cana-1326 190 30 2β€–d(x2nβˆ’1 2β€–d(x2nβˆ’1 NUM cana-1326 190 31 , , PUNCT cana-1326 190 32 x2n)β€– x2n)β€– PROPN cana-1326 190 33 ≀ ≀ PROPN cana-1326 190 34 β‹― β‹― PROPN cana-1326 190 35 ≀ ≀ NUM cana-1326 190 36 οΏ½ οΏ½ PROPN cana-1326 190 37 πœ† πœ† NOUN cana-1326 190 38 οΏ½ οΏ½ NOUN cana-1326 190 39 2𝑛+1β€–d(x0 2𝑛+1β€–d(x0 NUM cana-1326 190 40 , , PUNCT cana-1326 190 41 x1)β€– x1)β€– PROPN cana-1326 190 42 ( ( PUNCT cana-1326 190 43 3.8 3.8 NUM cana-1326 190 44 ) ) PUNCT cana-1326 190 45 that that PRON cana-1326 190 46 is, is, PROPN cana-1326 190 47 οΏ½ οΏ½ PROPN cana-1326 190 48 β€–d(xn+1 β€–d(xn+1 PROPN cana-1326 190 49 , , PUNCT cana-1326 190 50 xn+2)β€– xn+2)β€– PROPN cana-1326 190 51 ≀ ≀ PROPN cana-1326 190 52 οΏ½ οΏ½ PROPN cana-1326 190 53 πœ† πœ† NOUN cana-1326 190 54 οΏ½ οΏ½ PROPN cana-1326 190 55 β€–d(xn β€–d(xn PROPN cana-1326 190 56 , , PUNCT cana-1326 190 57 xn+1)β€– xn+1)β€– PROPN cana-1326 190 58 ≀ ≀ PROPN cana-1326 190 59 οΏ½ οΏ½ PROPN cana-1326 190 60 πœ† πœ† NOUN cana-1326 190 61 οΏ½ οΏ½ PROPN cana-1326 190 62 2β€–d(xnβˆ’1 2β€–d(xnβˆ’1 NUM cana-1326 190 63 , , PUNCT cana-1326 190 64 xn)β€– xn)β€– ADJ cana-1326 190 65 ≀ ≀ PROPN cana-1326 190 66 β‹― β‹― ADP cana-1326 190 67 ≀ ≀ NUM cana-1326 190 68 οΏ½ οΏ½ PROPN cana-1326 190 69 πœ† πœ† NOUN cana-1326 190 70 οΏ½ οΏ½ PROPN cana-1326 190 71 𝑛+1β€–d(x0 𝑛+1β€–d(x0 NOUN cana-1326 190 72 , , PUNCT cana-1326 190 73 x1)β€– x1)β€– PROPN cana-1326 190 74 ( ( PUNCT cana-1326 190 75 3.9 3.9 NUM cana-1326 190 76 ) ) PUNCT cana-1326 190 77 thus thus ADV cana-1326 190 78 , , PUNCT cana-1326 190 79 for for ADP cana-1326 190 80 any any DET cana-1326 190 81 m m NOUN cana-1326 190 82 > > X cana-1326 190 83 𝑛 𝑛 PROPN cana-1326 190 84 , , PUNCT cana-1326 190 85 m m PROPN cana-1326 190 86 , , PUNCT cana-1326 190 87 n n PROPN cana-1326 190 88 ∈ ∈ PROPN cana-1326 190 89 β„• β„• PROPN cana-1326 190 90 , , PUNCT cana-1326 190 91 we we PRON cana-1326 190 92 have have VERB cana-1326 190 93 οΏ½ οΏ½ NOUN cana-1326 190 94 β€–d(xn β€–d(xn NOUN cana-1326 190 95 , , PUNCT cana-1326 190 96 xm)β€– xm)β€– PROPN cana-1326 190 97 ≀ ≀ NUM cana-1326 190 98 𝑠‖d(xn 𝑠‖d(xn NUM cana-1326 190 99 , , PUNCT cana-1326 190 100 xn+1)β€– xn+1)β€– PROPN cana-1326 190 101 + + CCONJ cana-1326 190 102 𝑠‖d(xn+1 𝑠‖d(xn+1 PROPN cana-1326 190 103 , , PUNCT cana-1326 190 104 xm)β€– xm)β€– PROPN cana-1326 190 105 ≀ ≀ NUM cana-1326 190 106 𝑠‖d(xn 𝑠‖d(xn NUM cana-1326 190 107 , , PUNCT cana-1326 190 108 xn+1)β€– xn+1)β€– PROPN cana-1326 190 109 + + CCONJ cana-1326 190 110 𝑠2‖𝑑(π‘₯𝑛+1 𝑠2‖𝑑(π‘₯𝑛+1 ADJ cana-1326 190 111 , , PUNCT cana-1326 190 112 π‘₯𝑛+2β€– π‘₯𝑛+2β€– PROPN cana-1326 190 113 + + NOUN cana-1326 190 114 𝑠2‖𝑑(π‘₯𝑛+2 𝑠2‖𝑑(π‘₯𝑛+2 NUM cana-1326 190 115 , , PUNCT cana-1326 190 116 π‘₯π‘šβ€– π‘₯π‘šβ€– PROPN cana-1326 190 117 ≀ ≀ PROPN cana-1326 190 118 𝑠‖d(xn 𝑠‖d(xn PROPN cana-1326 190 119 , , PUNCT cana-1326 190 120 xn+1)β€– xn+1)β€– PROPN cana-1326 190 121 + + CCONJ cana-1326 190 122 𝑠2‖𝑑(π‘₯𝑛+1 𝑠2‖𝑑(π‘₯𝑛+1 ADJ cana-1326 190 123 , , PUNCT cana-1326 191 1 π‘₯𝑛+2β€– π‘₯𝑛+2β€– DET cana-1326 191 2 + + ADJ cana-1326 191 3 β‹―+ β‹―+ NOUN cana-1326 191 4 π‘ π‘šβˆ’π‘›βˆ’1‖𝑑(π‘₯π‘šβˆ’2 π‘ π‘šβˆ’π‘›βˆ’1‖𝑑(π‘₯π‘šβˆ’2 NOUN cana-1326 191 5 , , PUNCT cana-1326 191 6 π‘₯π‘šβˆ’1β€– π‘₯π‘šβˆ’1β€– PROPN cana-1326 191 7 communications communication NOUN cana-1326 191 8 on on ADP cana-1326 191 9 applied apply VERB cana-1326 191 10 nonlinear nonlinear ADJ cana-1326 191 11 analysis analysis NOUN cana-1326 191 12 issn issn NOUN cana-1326 191 13 : : PUNCT cana-1326 191 14 1074 1074 NUM cana-1326 191 15 - - PUNCT cana-1326 191 16 133x 133x NUM cana-1326 191 17 vol vol NOUN cana-1326 191 18 31 31 NUM cana-1326 191 19 no no NOUN cana-1326 191 20 . . PUNCT cana-1326 192 1 7s 7 NOUN cana-1326 192 2 ( ( PUNCT cana-1326 192 3 2024 2024 NUM cana-1326 192 4 ) ) PUNCT cana-1326 192 5 475 475 NUM cana-1326 192 6 https://internationalpubls.com https://internationalpubls.com X cana-1326 192 7 οΏ½ οΏ½ PROPN cana-1326 192 8 οΏ½ οΏ½ PROPN cana-1326 192 9 οΏ½ οΏ½ PROPN cana-1326 192 10 οΏ½ οΏ½ PROPN cana-1326 192 11 οΏ½ οΏ½ PROPN cana-1326 192 12 οΏ½ οΏ½ PROPN cana-1326 192 13 οΏ½ οΏ½ PROPN cana-1326 192 14 οΏ½ οΏ½ PROPN cana-1326 192 15 οΏ½ οΏ½ PROPN cana-1326 192 16 οΏ½ οΏ½ PROPN cana-1326 192 17 οΏ½ οΏ½ PROPN cana-1326 192 18 οΏ½ οΏ½ PROPN cana-1326 192 19 οΏ½ οΏ½ PROPN cana-1326 192 20 οΏ½ οΏ½ PROPN cana-1326 192 21 οΏ½ οΏ½ PROPN cana-1326 192 22 οΏ½ οΏ½ PROPN cana-1326 192 23 οΏ½ οΏ½ PROPN cana-1326 192 24 οΏ½ οΏ½ PROPN cana-1326 192 25 οΏ½ οΏ½ PROPN cana-1326 192 26 οΏ½ οΏ½ PROPN cana-1326 192 27 οΏ½ οΏ½ PROPN cana-1326 192 28 οΏ½ οΏ½ PROPN cana-1326 192 29 οΏ½ οΏ½ PROPN cana-1326 192 30 οΏ½ οΏ½ PROPN cana-1326 192 31 οΏ½ οΏ½ PROPN cana-1326 192 32 οΏ½ οΏ½ PROPN cana-1326 192 33 οΏ½ οΏ½ PROPN cana-1326 192 34 οΏ½ οΏ½ PROPN cana-1326 192 35 οΏ½ οΏ½ PROPN cana-1326 192 36 οΏ½ οΏ½ PROPN cana-1326 192 37 οΏ½ οΏ½ PROPN cana-1326 192 38 οΏ½ οΏ½ PROPN cana-1326 192 39 οΏ½ οΏ½ PROPN cana-1326 192 40 οΏ½ οΏ½ PROPN cana-1326 192 41 οΏ½ οΏ½ PROPN cana-1326 192 42 οΏ½ οΏ½ PROPN cana-1326 192 43 οΏ½ οΏ½ PROPN cana-1326 192 44 οΏ½ οΏ½ PROPN cana-1326 192 45 οΏ½ οΏ½ PROPN cana-1326 192 46 οΏ½ οΏ½ PROPN cana-1326 192 47 οΏ½ οΏ½ PROPN cana-1326 192 48 οΏ½ οΏ½ PROPN cana-1326 192 49 οΏ½ οΏ½ PROPN cana-1326 192 50 οΏ½ οΏ½ PROPN cana-1326 192 51 οΏ½ οΏ½ PROPN cana-1326 192 52 οΏ½ οΏ½ PROPN cana-1326 192 53 οΏ½ οΏ½ PROPN cana-1326 192 54 οΏ½ οΏ½ PROPN cana-1326 192 55 οΏ½ οΏ½ PROPN cana-1326 192 56 οΏ½ οΏ½ PROPN cana-1326 192 57 οΏ½ οΏ½ PROPN cana-1326 192 58 οΏ½ οΏ½ PROPN cana-1326 192 59 οΏ½ οΏ½ PROPN cana-1326 192 60 οΏ½ οΏ½ PROPN cana-1326 192 61 οΏ½ οΏ½ PROPN cana-1326 192 62 οΏ½ οΏ½ PROPN cana-1326 192 63 οΏ½ οΏ½ PROPN cana-1326 192 64 οΏ½ οΏ½ PROPN cana-1326 192 65 οΏ½ οΏ½ PROPN cana-1326 192 66 οΏ½ οΏ½ PROPN cana-1326 192 67 οΏ½ οΏ½ PROPN cana-1326 192 68 οΏ½ οΏ½ PROPN cana-1326 192 69 οΏ½ οΏ½ PROPN cana-1326 192 70 οΏ½ οΏ½ PROPN cana-1326 192 71 οΏ½ οΏ½ PROPN cana-1326 192 72 οΏ½ οΏ½ PROPN cana-1326 192 73 οΏ½ οΏ½ PROPN cana-1326 192 74 οΏ½ οΏ½ PROPN cana-1326 192 75 οΏ½ οΏ½ PROPN cana-1326 192 76 οΏ½ οΏ½ PROPN cana-1326 192 77 οΏ½ οΏ½ PROPN cana-1326 192 78 οΏ½ οΏ½ PROPN cana-1326 192 79 οΏ½ οΏ½ PROPN cana-1326 192 80 οΏ½ οΏ½ PROPN cana-1326 192 81 οΏ½ οΏ½ PROPN cana-1326 192 82 οΏ½ οΏ½ PROPN cana-1326 192 83 οΏ½ οΏ½ PROPN cana-1326 192 84 οΏ½ οΏ½ PROPN cana-1326 192 85 οΏ½ οΏ½ PROPN cana-1326 192 86 οΏ½ οΏ½ PROPN cana-1326 192 87 οΏ½ οΏ½ PROPN cana-1326 192 88 οΏ½ οΏ½ PROPN cana-1326 192 89 οΏ½ οΏ½ PROPN cana-1326 192 90 οΏ½ οΏ½ PROPN cana-1326 192 91 οΏ½ οΏ½ PROPN cana-1326 192 92 οΏ½ οΏ½ PROPN cana-1326 192 93 οΏ½ οΏ½ PROPN cana-1326 192 94 οΏ½ οΏ½ PROPN cana-1326 192 95 οΏ½ οΏ½ PROPN cana-1326 192 96 οΏ½ οΏ½ PROPN cana-1326 192 97 οΏ½ οΏ½ PROPN cana-1326 192 98 οΏ½ οΏ½ PROPN cana-1326 192 99 οΏ½ οΏ½ PROPN cana-1326 192 100 οΏ½ οΏ½ PROPN cana-1326 192 101 οΏ½ οΏ½ PROPN cana-1326 192 102 οΏ½ οΏ½ PROPN cana-1326 192 103 οΏ½ οΏ½ PROPN cana-1326 192 104 οΏ½ οΏ½ PROPN cana-1326 192 105 οΏ½ οΏ½ PROPN cana-1326 192 106 οΏ½ οΏ½ PROPN cana-1326 192 107 οΏ½ οΏ½ PROPN cana-1326 192 108 οΏ½ οΏ½ PROPN cana-1326 192 109 οΏ½ οΏ½ PROPN cana-1326 192 110 οΏ½ οΏ½ PROPN cana-1326 192 111 οΏ½ οΏ½ PROPN cana-1326 192 112 οΏ½ οΏ½ PROPN cana-1326 192 113 οΏ½ οΏ½ PROPN cana-1326 192 114 οΏ½ οΏ½ PROPN cana-1326 192 115 οΏ½ οΏ½ PROPN cana-1326 192 116 οΏ½ οΏ½ PROPN cana-1326 192 117 οΏ½ οΏ½ PROPN cana-1326 192 118 οΏ½ οΏ½ PROPN cana-1326 192 119 οΏ½ οΏ½ PROPN cana-1326 192 120 οΏ½ οΏ½ PROPN cana-1326 192 121 οΏ½ οΏ½ PROPN cana-1326 192 122 οΏ½ οΏ½ PROPN cana-1326 192 123 οΏ½ οΏ½ PROPN cana-1326 192 124 οΏ½ οΏ½ PROPN cana-1326 192 125 οΏ½ οΏ½ PROPN cana-1326 192 126 οΏ½ οΏ½ PROPN cana-1326 192 127 οΏ½ οΏ½ PROPN cana-1326 192 128 οΏ½ οΏ½ PROPN cana-1326 192 129 οΏ½ οΏ½ PROPN cana-1326 192 130 + + NOUN cana-1326 192 131 π‘ π‘šβˆ’π‘›β€–π‘‘(π‘₯π‘šβˆ’1 π‘ π‘šβˆ’π‘›β€–π‘‘(π‘₯π‘šβˆ’1 PROPN cana-1326 192 132 , , PUNCT cana-1326 192 133 π‘₯π‘šβ€– π‘₯π‘šβ€– PROPN cana-1326 192 134 ( ( PUNCT cana-1326 192 135 3.10 3.10 NUM cana-1326 192 136 ) ) PUNCT cana-1326 192 137 by by ADP cana-1326 192 138 using use VERB cana-1326 192 139 ( ( PUNCT cana-1326 192 140 3.9 3.9 NUM cana-1326 192 141 ) ) PUNCT cana-1326 192 142 we we PRON cana-1326 192 143 get get VERB cana-1326 192 144 , , PUNCT cana-1326 192 145 οΏ½ οΏ½ PROPN cana-1326 192 146 β€–d(xn β€–d(xn PROPN cana-1326 192 147 , , PUNCT cana-1326 192 148 xm)β€– xm)β€– PROPN cana-1326 192 149 ≀ ≀ PROPN cana-1326 192 150 π‘ πœ†π‘›β€–π‘‘(π‘₯0 π‘ πœ†π‘›β€–π‘‘(π‘₯0 ADJ cana-1326 192 151 , , PUNCT cana-1326 192 152 π‘₯1β€– π‘₯1β€– PROPN cana-1326 192 153 + + NUM cana-1326 192 154 𝑠2πœ†π‘›+1‖𝑑(π‘₯0 𝑠2πœ†π‘›+1‖𝑑(π‘₯0 ADJ cana-1326 192 155 , , PUNCT cana-1326 192 156 π‘₯1β€– π‘₯1β€– PROPN cana-1326 192 157 οΏ½ οΏ½ PROPN cana-1326 192 158 οΏ½ οΏ½ PROPN cana-1326 192 159 οΏ½ οΏ½ PROPN cana-1326 192 160 οΏ½ οΏ½ PROPN cana-1326 192 161 οΏ½ οΏ½ PROPN cana-1326 192 162 οΏ½ οΏ½ PROPN cana-1326 192 163 οΏ½ οΏ½ PROPN cana-1326 192 164 οΏ½ οΏ½ PROPN cana-1326 192 165 οΏ½ οΏ½ PROPN cana-1326 192 166 οΏ½ οΏ½ PROPN cana-1326 192 167 οΏ½ οΏ½ PROPN cana-1326 192 168 οΏ½ οΏ½ PROPN cana-1326 192 169 οΏ½ οΏ½ PROPN cana-1326 192 170 οΏ½ οΏ½ PROPN cana-1326 192 171 οΏ½ οΏ½ PROPN cana-1326 192 172 οΏ½ οΏ½ PROPN cana-1326 192 173 οΏ½ οΏ½ PROPN cana-1326 192 174 οΏ½ οΏ½ PROPN cana-1326 192 175 οΏ½ οΏ½ PROPN cana-1326 192 176 οΏ½ οΏ½ PROPN cana-1326 192 177 οΏ½ οΏ½ PROPN cana-1326 192 178 οΏ½ οΏ½ PROPN cana-1326 192 179 οΏ½ οΏ½ PROPN cana-1326 192 180 οΏ½ οΏ½ PROPN cana-1326 192 181 οΏ½ οΏ½ PROPN cana-1326 192 182 οΏ½ οΏ½ PROPN cana-1326 192 183 οΏ½ οΏ½ PROPN cana-1326 192 184 οΏ½ οΏ½ PROPN cana-1326 192 185 οΏ½ οΏ½ PROPN cana-1326 192 186 οΏ½ οΏ½ PROPN cana-1326 192 187 οΏ½ οΏ½ PROPN cana-1326 192 188 οΏ½ οΏ½ PROPN cana-1326 192 189 οΏ½ οΏ½ PROPN cana-1326 192 190 οΏ½ οΏ½ PROPN cana-1326 192 191 οΏ½ οΏ½ PROPN cana-1326 192 192 οΏ½ οΏ½ PROPN cana-1326 192 193 + + NOUN cana-1326 192 194 𝑠3πœ†π‘›+2‖𝑑(π‘₯0 𝑠3πœ†π‘›+2‖𝑑(π‘₯0 ADJ cana-1326 192 195 , , PUNCT cana-1326 192 196 π‘₯1β€– π‘₯1β€– VERB cana-1326 192 197 + + ADJ cana-1326 192 198 β‹―+ β‹―+ NOUN cana-1326 192 199 οΏ½ οΏ½ NOUN cana-1326 192 200 π‘ π‘šβˆ’π‘›βˆ’1πœ†π‘šβˆ’2‖𝑑(π‘₯0 π‘ π‘šβˆ’π‘›βˆ’1πœ†π‘šβˆ’2‖𝑑(π‘₯0 VERB cana-1326 192 201 , , PUNCT cana-1326 192 202 π‘₯1β€– π‘₯1β€– PROPN cana-1326 192 203 + + NUM cana-1326 192 204 π‘ π‘šβˆ’π‘›πœ†π‘šβˆ’1‖𝑑(π‘₯0 π‘ π‘šβˆ’π‘›πœ†π‘šβˆ’1‖𝑑(π‘₯0 ADJ cana-1326 192 205 , , PUNCT cana-1326 192 206 π‘₯1β€– π‘₯1β€– PROPN cana-1326 192 207 = = PUNCT cana-1326 192 208 βˆ‘ βˆ‘ ADP cana-1326 192 209 π‘ π‘–πœ†π‘›+π‘–βˆ’1π‘šβˆ’π‘› π‘ π‘–πœ†π‘›+π‘–βˆ’1π‘šβˆ’π‘› ADJ cana-1326 192 210 𝑖=1 𝑖=1 PROPN cana-1326 192 211 ‖𝑑(π‘₯0 ‖𝑑(π‘₯0 PROPN cana-1326 192 212 , , PUNCT cana-1326 192 213 π‘₯1β€– π‘₯1β€– PROPN cana-1326 192 214 ≀ ≀ NUM cana-1326 192 215 π‘ πœ†π‘› π‘ πœ†π‘› NOUN cana-1326 192 216 1βˆ’πœ†π‘  1βˆ’πœ†π‘  NUM cana-1326 192 217 ‖𝑑(π‘₯0 ‖𝑑(π‘₯0 PROPN cana-1326 192 218 , , PUNCT cana-1326 192 219 π‘₯1β€– π‘₯1β€– PROPN cana-1326 192 220 β†’0 β†’0 PUNCT cana-1326 192 221 as as ADP cana-1326 192 222 m m PROPN cana-1326 192 223 , , PUNCT cana-1326 192 224 nβ†’βˆž nβ†’βˆž X cana-1326 192 225 ( ( PUNCT cana-1326 192 226 3.11 3.11 NUM cana-1326 192 227 ) ) PUNCT cana-1326 192 228 this this PRON cana-1326 192 229 implies imply VERB cana-1326 192 230 that that SCONJ cana-1326 192 231 the the DET cana-1326 192 232 sequence sequence NOUN cana-1326 192 233 { { PUNCT cana-1326 192 234 π‘₯𝑛+ π‘₯𝑛+ PROPN cana-1326 192 235 οΏ½ οΏ½ PROPN cana-1326 192 236 π‘Žπ‘  π‘Žπ‘  PROPN cana-1326 192 237 οΏ½ οΏ½ PROPN cana-1326 192 238 a a DET cana-1326 192 239 cauchy cauchy ADJ cana-1326 192 240 sequence sequence NOUN cana-1326 192 241 in in ADP cana-1326 192 242 𝑋.since 𝑋.since NOUN cana-1326 192 243 x x PRON cana-1326 192 244 is be AUX cana-1326 192 245 complete complete ADJ cana-1326 192 246 , , PUNCT cana-1326 192 247 there there PRON cana-1326 192 248 exists exist VERB cana-1326 192 249 a a DET cana-1326 192 250 point point NOUN cana-1326 192 251 u∈ u∈ ADJ cana-1326 192 252 𝑋 𝑋 NOUN cana-1326 192 253 with with ADP cana-1326 192 254 limπ‘›β†’βˆž limπ‘›β†’βˆž PROPN cana-1326 192 255 π‘₯𝑛 π‘₯𝑛 PROPN cana-1326 192 256 = = SYM cana-1326 192 257 𝑒. 𝑒. NOUN cana-1326 192 258 assume assume VERB cana-1326 192 259 not not PART cana-1326 192 260 , , PUNCT cana-1326 192 261 then then ADV cana-1326 192 262 there there PRON cana-1326 192 263 exists exist VERB cana-1326 192 264 v∈ v∈ PROPN cana-1326 192 265 𝑋 𝑋 PROPN cana-1326 192 266 such such ADJ cana-1326 192 267 that that SCONJ cana-1326 192 268 ‖𝑑(𝑒 ‖𝑑(𝑒 VERB cana-1326 192 269 , , PUNCT cana-1326 192 270 𝑆𝑒)β€– 𝑆𝑒)β€– ADJ cana-1326 192 271 = = ADJ cana-1326 192 272 ||v||>0 ||v||>0 NOUN cana-1326 192 273 ( ( PUNCT cana-1326 192 274 3.12 3.12 NUM cana-1326 192 275 ) ) PUNCT cana-1326 192 276 so so ADV cana-1326 192 277 by by ADP cana-1326 192 278 using use VERB cana-1326 192 279 the the DET cana-1326 192 280 triangular triangular NOUN cana-1326 192 281 inequality inequality NOUN cana-1326 192 282 and and CCONJ cana-1326 192 283 ( ( PUNCT cana-1326 192 284 3.1 3.1 NUM cana-1326 192 285 ) ) PUNCT cana-1326 192 286 , , PUNCT cana-1326 192 287 we we PRON cana-1326 192 288 get get VERB cana-1326 192 289 v v ADP cana-1326 192 290 = = SYM cana-1326 192 291 d(u d(u PROPN cana-1326 192 292 , , PUNCT cana-1326 192 293 su) su) PROPN cana-1326 192 294 οΏ½ οΏ½ PROPN cana-1326 192 295 ≲𝑖2s ≲𝑖2s PROPN cana-1326 192 296 οΏ½ οΏ½ PROPN cana-1326 192 297 𝑑(𝑒 𝑑(𝑒 NOUN cana-1326 192 298 , , PUNCT cana-1326 192 299 π‘₯2𝑛+2)+s π‘₯2𝑛+2)+s X cana-1326 192 300 οΏ½ οΏ½ X cana-1326 192 301 𝑑(π‘₯2𝑛+2 𝑑(π‘₯2𝑛+2 NOUN cana-1326 192 302 , , PUNCT cana-1326 192 303 𝑆𝑒 𝑆𝑒 NOUN cana-1326 192 304 ) ) PUNCT cana-1326 192 305 ≲𝑖2 ≲𝑖2 PROPN cana-1326 192 306 s s PROPN cana-1326 192 307 οΏ½ οΏ½ NOUN cana-1326 192 308 𝑑(𝑒 𝑑(𝑒 ADJ cana-1326 192 309 , , PUNCT cana-1326 192 310 π‘₯2𝑛+2)+s π‘₯2𝑛+2)+s X cana-1326 192 311 οΏ½ οΏ½ NOUN cana-1326 192 312 𝑑(𝑇π‘₯2𝑛+1 𝑑(𝑇π‘₯2𝑛+1 NOUN cana-1326 192 313 , , PUNCT cana-1326 192 314 𝑆𝑒 𝑆𝑒 PROPN cana-1326 192 315 ) ) PUNCT cana-1326 192 316 ≲𝑖2 ≲𝑖2 PROPN cana-1326 192 317 s s PROPN cana-1326 192 318 οΏ½ οΏ½ NOUN cana-1326 192 319 𝑑(𝑒 𝑑(𝑒 ADJ cana-1326 192 320 , , PUNCT cana-1326 192 321 π‘₯2𝑛+2)+s π‘₯2𝑛+2)+s X cana-1326 192 322 οΏ½ οΏ½ NOUN cana-1326 192 323 𝐴𝑑(𝑒 𝐴𝑑(𝑒 NOUN cana-1326 192 324 , , PUNCT cana-1326 192 325 π‘₯2𝑛+1)+ π‘₯2𝑛+1)+ ADJ cana-1326 192 326 οΏ½ οΏ½ PROPN cana-1326 192 327 𝑠𝐡 𝑠𝐡 PROPN cana-1326 192 328 d(u d(u PROPN cana-1326 192 329 , , PUNCT cana-1326 192 330 su)d(x2n+1,tx2n+1 su)d(x2n+1,tx2n+1 ADJ cana-1326 192 331 ) ) PUNCT cana-1326 192 332 d(u d(u PROPN cana-1326 192 333 , , PUNCT cana-1326 192 334 tx2n+1)+𝑑(x2n+1,su)+d(u tx2n+1)+𝑑(x2n+1,su)+d(u PROPN cana-1326 192 335 , , PUNCT cana-1326 192 336 x2n+1 x2n+1 PROPN cana-1326 192 337 ) ) PUNCT cana-1326 192 338 ( ( PUNCT cana-1326 192 339 3.13 3.13 NUM cana-1326 192 340 ) ) PUNCT cana-1326 192 341 which which PRON cana-1326 192 342 implies imply VERB cana-1326 192 343 that that SCONJ cana-1326 192 344 ‖𝑣‖ ‖𝑣‖ PROPN cana-1326 192 345 = = SYM cana-1326 192 346 οΏ½ οΏ½ PROPN cana-1326 192 347 ‖𝑑(𝑒 ‖𝑑(𝑒 NOUN cana-1326 192 348 , , PUNCT cana-1326 192 349 𝑆𝑒)β€– 𝑆𝑒)β€– ADJ cana-1326 192 350 ≀ ≀ NOUN cana-1326 192 351 𝑠‖𝑑(𝑒 𝑠‖𝑑(𝑒 NOUN cana-1326 192 352 , , PUNCT cana-1326 192 353 π‘₯2𝑛+2)β€– π‘₯2𝑛+2)β€– PROPN cana-1326 192 354 + + CCONJ cana-1326 192 355 𝐴𝑠‖𝑑(𝑒 𝐴𝑠‖𝑑(𝑒 ADJ cana-1326 192 356 , , PUNCT cana-1326 192 357 π‘₯2𝑛+1)β€– π‘₯2𝑛+1)β€– ADJ cana-1326 192 358 + + CCONJ cana-1326 192 359 π‘ π΅βˆš2 π‘ π΅βˆš2 NOUN cana-1326 192 360 ‖𝑑(π‘₯2𝑛+1,π‘₯2𝑛+2)‖‖𝑑(𝑒,𝑆𝑒)β€– ‖𝑑(π‘₯2𝑛+1,π‘₯2𝑛+2)‖‖𝑑(𝑒,𝑆𝑒)β€– NOUN cana-1326 192 361 β€–d(u β€–d(u ADV cana-1326 192 362 , , PUNCT cana-1326 192 363 tx2n+1)β€–+‖𝑑(π‘₯2𝑛+1,𝑆𝑒)β€–+‖𝑑(𝑒,π‘₯2𝑛+1)β€– tx2n+1)β€–+‖𝑑(π‘₯2𝑛+1,𝑆𝑒)β€–+‖𝑑(𝑒,π‘₯2𝑛+1)β€– NOUN cana-1326 192 364 taking taking NOUN cana-1326 192 365 limit limit NOUN cana-1326 192 366 as as ADP cana-1326 192 367 nβ†’ nβ†’ PROPN cana-1326 192 368 ∞ ∞ PROPN cana-1326 192 369 , , PUNCT cana-1326 192 370 we we PRON cana-1326 192 371 get get VERB cana-1326 192 372 ‖𝑣‖ ‖𝑣‖ PROPN cana-1326 192 373 = = SYM cana-1326 192 374 οΏ½ οΏ½ PROPN cana-1326 192 375 ‖𝑑(𝑒 ‖𝑑(𝑒 NOUN cana-1326 192 376 , , PUNCT cana-1326 192 377 𝑆𝑒)β€– 𝑆𝑒)β€– VERB cana-1326 192 378 ≀ ≀ NOUN cana-1326 192 379 0 0 NUM cana-1326 192 380 , , PUNCT cana-1326 192 381 a a DET cana-1326 192 382 contradiction contradiction NOUN cana-1326 192 383 with with ADP cana-1326 192 384 ( ( PUNCT cana-1326 192 385 3.12 3.12 NUM cana-1326 192 386 ) ) PUNCT cana-1326 192 387 . . PUNCT cana-1326 193 1 so so ADV cana-1326 193 2 ‖𝑣‖ ‖𝑣‖ ADJ cana-1326 193 3 = = SYM cana-1326 193 4 0 0 PUNCT cana-1326 194 1 hence hence ADV cana-1326 194 2 ‖𝑑(𝑒 ‖𝑑(𝑒 NOUN cana-1326 194 3 , , PUNCT cana-1326 194 4 𝑆𝑒)β€– 𝑆𝑒)β€– ADJ cana-1326 194 5 = = NOUN cana-1326 194 6 0 0 NUM cana-1326 194 7 that that PRON cana-1326 194 8 is be AUX cana-1326 194 9 u u NOUN cana-1326 194 10 = = PROPN cana-1326 194 11 su su PROPN cana-1326 194 12 . . PUNCT cana-1326 195 1 similarly similarly ADV cana-1326 195 2 , , PUNCT cana-1326 195 3 we we PRON cana-1326 195 4 obtain obtain VERB cana-1326 195 5 u u NOUN cana-1326 195 6 = = PROPN cana-1326 195 7 tu tu PROPN cana-1326 195 8 . . PROPN cana-1326 196 1 for for ADP cana-1326 196 2 uniqueness uniqueness NOUN cana-1326 196 3 assume assume VERB cana-1326 196 4 that that SCONJ cana-1326 196 5 π‘’βˆ— π‘’βˆ— NOUN cana-1326 196 6 in in ADP cana-1326 196 7 𝑋 𝑋 PROPN cana-1326 196 8 οΏ½ οΏ½ PROPN cana-1326 196 9 is be AUX cana-1326 196 10 another another DET cana-1326 196 11 common common ADJ cana-1326 196 12 fixed fix VERB cana-1326 196 13 point point NOUN cana-1326 196 14 of of ADP cana-1326 196 15 s s PRON cana-1326 196 16 and and CCONJ cana-1326 196 17 t. t. NOUN cana-1326 196 18 then then ADV cana-1326 196 19 𝑑(𝑒 𝑑(𝑒 NOUN cana-1326 196 20 , , PUNCT cana-1326 196 21 π‘’βˆ— π‘’βˆ— NOUN cana-1326 196 22 ) ) PUNCT cana-1326 197 1 = = SYM cana-1326 197 2 𝑑(𝑆𝑒 𝑑(𝑆𝑒 NOUN cana-1326 197 3 , , PUNCT cana-1326 197 4 π‘‡π‘’βˆ— π‘‡π‘’βˆ— NOUN cana-1326 197 5 ) ) PUNCT cana-1326 197 6 ≲𝑖2 ≲𝑖2 PROPN cana-1326 197 7 𝐴 𝐴 PROPN cana-1326 197 8 οΏ½ οΏ½ PROPN cana-1326 197 9 𝑑(𝑒 𝑑(𝑒 PROPN cana-1326 197 10 , , PUNCT cana-1326 197 11 𝑒 𝑒 PRON cana-1326 197 12 βˆ— βˆ— NOUN cana-1326 197 13 ) ) PUNCT cana-1326 197 14 + + CCONJ cana-1326 197 15 d(u d(u PROPN cana-1326 197 16 , , PUNCT cana-1326 197 17 su)d(π‘’βˆ— su)d(π‘’βˆ— ADJ cana-1326 197 18 , , PUNCT cana-1326 197 19 π‘‡π‘’βˆ— π‘‡π‘’βˆ— NOUN cana-1326 197 20 ) ) PUNCT cana-1326 197 21 d(𝑒 d(𝑒 PROPN cana-1326 197 22 , , PUNCT cana-1326 197 23 π‘‡π‘’βˆ— π‘‡π‘’βˆ— NOUN cana-1326 197 24 ) ) PUNCT cana-1326 197 25 + + SYM cana-1326 197 26 𝑑(π‘’βˆ— 𝑑(π‘’βˆ— NOUN cana-1326 197 27 , , PUNCT cana-1326 197 28 𝑆𝑒 𝑆𝑒 PROPN cana-1326 197 29 ) ) PUNCT cana-1326 197 30 + + PUNCT cana-1326 197 31 d(𝑒 d(𝑒 PROPN cana-1326 197 32 , , PUNCT cana-1326 197 33 π‘’βˆ— π‘’βˆ— NOUN cana-1326 197 34 ) ) PUNCT cana-1326 198 1 so so SCONJ cana-1326 198 2 that that SCONJ cana-1326 198 3 ‖𝑑(𝑒 ‖𝑑(𝑒 ADP cana-1326 198 4 , , PUNCT cana-1326 198 5 π‘’βˆ—)β€– π‘’βˆ—)β€– ADJ cana-1326 198 6 = = SYM cana-1326 198 7 ‖𝑑(𝑆𝑒 ‖𝑑(𝑆𝑒 NOUN cana-1326 198 8 , , PUNCT cana-1326 198 9 π‘‡π‘’βˆ—)β€– π‘‡π‘’βˆ—)β€– PROPN cana-1326 198 10 ≀ ≀ PUNCT cana-1326 198 11 a‖𝑑(𝑒 a‖𝑑(𝑒 PROPN cana-1326 198 12 , , PUNCT cana-1326 198 13 π‘’βˆ—)β€– π‘’βˆ—)β€– PUNCT cana-1326 198 14 + + PUNCT cana-1326 198 15 𝐡√2 𝐡√2 PROPN cana-1326 198 16 ‖𝑑(𝑒,𝑆𝑒)‖‖𝑑(π‘’βˆ—,π‘‡π‘’βˆ—)β€– ‖𝑑(𝑒,𝑆𝑒)‖‖𝑑(π‘’βˆ—,π‘‡π‘’βˆ—)β€– PROPN cana-1326 198 17 ‖𝑑(𝑒,π‘‡π‘’βˆ—)β€–+‖𝑑(π‘’βˆ—,𝑆𝑒)β€–+‖𝑑(𝑒,π‘’βˆ—)β€– ‖𝑑(𝑒,π‘‡π‘’βˆ—)β€–+‖𝑑(π‘’βˆ—,𝑆𝑒)β€–+‖𝑑(𝑒,π‘’βˆ—)β€– PUNCT cana-1326 198 18 ≀ ≀ PROPN cana-1326 198 19 𝐴‖𝑑(𝑒 𝐴‖𝑑(𝑒 NOUN cana-1326 198 20 , , PUNCT cana-1326 198 21 π‘’βˆ—)β€– π‘’βˆ—)β€– X cana-1326 198 22 hence hence ADV cana-1326 198 23 𝑒 𝑒 X cana-1326 198 24 = = X cana-1326 198 25 π‘’βˆ—. π‘’βˆ—. X cana-1326 198 26 therefore therefore ADV cana-1326 198 27 u u NOUN cana-1326 198 28 is be AUX cana-1326 198 29 a a DET cana-1326 198 30 unique unique ADJ cana-1326 198 31 common common ADJ cana-1326 198 32 fixed fix VERB cana-1326 198 33 point point NOUN cana-1326 198 34 of of ADP cana-1326 198 35 t t PROPN cana-1326 198 36 and and CCONJ cana-1326 198 37 s. s. PROPN cana-1326 198 38 now now ADV cana-1326 198 39 , , PUNCT cana-1326 198 40 we we PRON cana-1326 198 41 consider consider VERB cana-1326 198 42 the the DET cana-1326 198 43 second second ADJ cana-1326 198 44 case case NOUN cana-1326 198 45 : : PUNCT cana-1326 198 46 𝑑 𝑑 PROPN cana-1326 198 47 οΏ½ οΏ½ PROPN cana-1326 198 48 (π‘₯ (π‘₯ PROPN cana-1326 198 49 οΏ½ οΏ½ PROPN cana-1326 198 50 , , PUNCT cana-1326 198 51 𝑇 𝑇 PROPN cana-1326 198 52 οΏ½ οΏ½ PROPN cana-1326 198 53 𝑦 𝑦 NOUN cana-1326 198 54 οΏ½ οΏ½ PROPN cana-1326 198 55 ) ) PUNCT cana-1326 198 56 + + CCONJ cana-1326 198 57 𝑑 𝑑 PROPN cana-1326 198 58 οΏ½ οΏ½ PROPN cana-1326 198 59 (𝑦 (𝑦 NOUN cana-1326 198 60 οΏ½ οΏ½ PROPN cana-1326 198 61 , , PUNCT cana-1326 198 62 𝑆 𝑆 PROPN cana-1326 198 63 οΏ½ οΏ½ PROPN cana-1326 198 64 π‘₯ π‘₯ NOUN cana-1326 198 65 οΏ½ οΏ½ PROPN cana-1326 198 66 ) ) PUNCT cana-1326 198 67 + + CCONJ cana-1326 198 68 𝑑 𝑑 PROPN cana-1326 198 69 οΏ½ οΏ½ PROPN cana-1326 198 70 (π‘₯ (π‘₯ PROPN cana-1326 198 71 οΏ½ οΏ½ PROPN cana-1326 198 72 , , PUNCT cana-1326 198 73 𝑦 𝑦 NOUN cana-1326 198 74 οΏ½ οΏ½ NOUN cana-1326 198 75 ) ) PUNCT cana-1326 198 76 = = SYM cana-1326 199 1 0 0 X cana-1326 199 2 . . PUNCT cana-1326 200 1 put put VERB cana-1326 200 2 x= x= PROPN cana-1326 200 3 οΏ½ οΏ½ PROPN cana-1326 200 4 x2n x2n PROPN cana-1326 200 5 and and CCONJ cana-1326 200 6 y y PROPN cana-1326 200 7 = = PROPN cana-1326 200 8 x2n+1 x2n+1 PROPN cana-1326 200 9 in in ADP cana-1326 200 10 this this DET cana-1326 200 11 expression expression NOUN cana-1326 200 12 we we PRON cana-1326 200 13 get get VERB cana-1326 200 14 ( ( PUNCT cana-1326 200 15 x2n x2n ADV cana-1326 200 16 , , PUNCT cana-1326 200 17 𝑇 𝑇 PROPN cana-1326 200 18 οΏ½ οΏ½ PROPN cana-1326 200 19 x2n+1 x2n+1 PROPN cana-1326 200 20 ) ) PUNCT cana-1326 201 1 + + CCONJ cana-1326 201 2 ( ( PUNCT cana-1326 201 3 x2n+1 x2n+1 PROPN cana-1326 201 4 , , PUNCT cana-1326 201 5 𝑆 𝑆 PROPN cana-1326 201 6 οΏ½ οΏ½ PROPN cana-1326 201 7 x2n x2n PROPN cana-1326 201 8 ) ) PUNCT cana-1326 201 9 + + CCONJ cana-1326 201 10 ( ( PUNCT cana-1326 201 11 x2n, x2n, PROPN cana-1326 201 12 οΏ½ οΏ½ PROPN cana-1326 201 13 x2n+1 x2n+1 PROPN cana-1326 201 14 ) ) PUNCT cana-1326 202 1 = = SYM cana-1326 202 2 0 0 PUNCT cana-1326 202 3 ( ( PUNCT cana-1326 202 4 for for ADP cana-1326 202 5 any any DET cana-1326 202 6 n n NOUN cana-1326 202 7 ) ) PUNCT cana-1326 202 8 which which PRON cana-1326 202 9 implies imply VERB cana-1326 202 10 ( ( PUNCT cana-1326 202 11 sx2n sx2n PROPN cana-1326 202 12 , , PUNCT cana-1326 202 13 𝑇 𝑇 PROPN cana-1326 202 14 οΏ½ οΏ½ PROPN cana-1326 202 15 x2n+1)=0 x2n+1)=0 PROPN cana-1326 202 16 so so SCONJ cana-1326 202 17 that that SCONJ cana-1326 202 18 x2n x2n PUNCT cana-1326 202 19 = = PUNCT cana-1326 202 20 sx2n= sx2n= PROPN cana-1326 202 21 x2n+1 x2n+1 X cana-1326 202 22 = = PUNCT cana-1326 203 1 tx2n+1= tx2n+1= PROPN cana-1326 204 1 x2n+2 x2n+2 PROPN cana-1326 204 2 . . PUNCT cana-1326 205 1 thus thus ADV cana-1326 205 2 we we PRON cana-1326 205 3 have have VERB cana-1326 205 4 x2n x2n NOUN cana-1326 205 5 = = PUNCT cana-1326 205 6 sx2n= sx2n= PROPN cana-1326 205 7 x2n+1 x2n+1 PROPN cana-1326 205 8 , , PUNCT cana-1326 205 9 communications communication NOUN cana-1326 205 10 on on ADP cana-1326 205 11 applied apply VERB cana-1326 205 12 nonlinear nonlinear ADJ cana-1326 205 13 analysis analysis NOUN cana-1326 205 14 issn issn NOUN cana-1326 205 15 : : PUNCT cana-1326 205 16 1074 1074 NUM cana-1326 205 17 - - PUNCT cana-1326 205 18 133x 133x NUM cana-1326 205 19 vol vol NOUN cana-1326 205 20 31 31 NUM cana-1326 205 21 no no NOUN cana-1326 205 22 . . PUNCT cana-1326 206 1 7s 7 NOUN cana-1326 206 2 ( ( PUNCT cana-1326 206 3 2024 2024 NUM cana-1326 206 4 ) ) PUNCT cana-1326 206 5 476 476 NUM cana-1326 206 6 https://internationalpubls.com https://internationalpubls.com X cana-1326 207 1 so so ADV cana-1326 207 2 there there PRON cana-1326 207 3 exist exist VERB cana-1326 207 4 k1 k1 PROPN cana-1326 207 5 οΏ½ οΏ½ PROPN cana-1326 207 6 and and CCONJ cana-1326 207 7 l1 l1 PROPN cana-1326 207 8 such such ADJ cana-1326 207 9 that that SCONJ cana-1326 207 10 k1 k1 PROPN cana-1326 207 11 οΏ½ οΏ½ PROPN cana-1326 207 12 = = SYM cana-1326 207 13 sl1 sl1 PROPN cana-1326 207 14 = = SYM cana-1326 207 15 l1 l1 PROPN cana-1326 207 16 where where SCONJ cana-1326 207 17 k1 k1 PROPN cana-1326 207 18 = = SYM cana-1326 207 19 x2n+1 x2n+1 PROPN cana-1326 207 20 and and CCONJ cana-1326 207 21 l1 l1 PROPN cana-1326 207 22 = = PUNCT cana-1326 208 1 x2n x2n PROPN cana-1326 208 2 . . PUNCT cana-1326 209 1 using use VERB cana-1326 209 2 foregoing forego VERB cana-1326 209 3 arguments argument NOUN cana-1326 209 4 , , PUNCT cana-1326 209 5 one one PRON cana-1326 209 6 can can AUX cana-1326 209 7 also also ADV cana-1326 209 8 show show VERB cana-1326 209 9 that that SCONJ cana-1326 209 10 there there PRON cana-1326 209 11 exists exist VERB cana-1326 209 12 k2 k2 PROPN cana-1326 209 13 οΏ½ οΏ½ PROPN cana-1326 209 14 and and CCONJ cana-1326 209 15 l2 l2 VERB cana-1326 209 16 such such ADJ cana-1326 209 17 that that SCONJ cana-1326 209 18 k2 k2 PROPN cana-1326 209 19 οΏ½ οΏ½ PROPN cana-1326 209 20 = = SYM cana-1326 209 21 sl2 sl2 PROPN cana-1326 209 22 = = PUNCT cana-1326 209 23 l2 l2 NOUN cana-1326 209 24 where where SCONJ cana-1326 209 25 k2 k2 PROPN cana-1326 209 26 = = SYM cana-1326 209 27 x2n+2 x2n+2 PROPN cana-1326 209 28 and and CCONJ cana-1326 209 29 l2 l2 PROPN cana-1326 209 30 = = SYM cana-1326 209 31 x2n+1 x2n+1 PROPN cana-1326 209 32 . . PUNCT cana-1326 210 1 as as ADP cana-1326 210 2 ( ( PUNCT cana-1326 210 3 l1 l1 PROPN cana-1326 210 4 , , PUNCT cana-1326 210 5 𝑇 𝑇 PROPN cana-1326 210 6 οΏ½ οΏ½ PROPN cana-1326 210 7 l2 l2 NOUN cana-1326 210 8 ) ) PUNCT cana-1326 211 1 + + CCONJ cana-1326 211 2 ( ( PUNCT cana-1326 211 3 l2 l2 NOUN cana-1326 211 4 , , PUNCT cana-1326 211 5 𝑆 𝑆 PROPN cana-1326 211 6 οΏ½ οΏ½ PROPN cana-1326 211 7 l1 l1 PROPN cana-1326 211 8 ) ) PUNCT cana-1326 212 1 + + CCONJ cana-1326 212 2 ( ( PUNCT cana-1326 212 3 l1 l1 PROPN cana-1326 212 4 , , PUNCT cana-1326 212 5 l2 l2 NOUN cana-1326 212 6 ) ) PUNCT cana-1326 212 7 = = SYM cana-1326 212 8 0 0 PUNCT cana-1326 212 9 ( ( PUNCT cana-1326 212 10 from from ADP cana-1326 212 11 definition definition NOUN cana-1326 212 12 ) ) PUNCT cana-1326 212 13 implies imply VERB cana-1326 212 14 𝑑 𝑑 PROPN cana-1326 212 15 οΏ½ οΏ½ PROPN cana-1326 212 16 (sl1 (sl1 NOUN cana-1326 212 17 , , PUNCT cana-1326 212 18 𝑇 𝑇 PROPN cana-1326 212 19 οΏ½ οΏ½ PROPN cana-1326 212 20 l2)=0,therefore l2)=0,therefore NOUN cana-1326 212 21 k1 k1 X cana-1326 212 22 οΏ½ οΏ½ PROPN cana-1326 212 23 = = SYM cana-1326 212 24 sl1 sl1 PROPN cana-1326 212 25 = = SYM cana-1326 212 26 tl2 tl2 NOUN cana-1326 212 27 = = SYM cana-1326 212 28 k2 k2 NOUN cana-1326 212 29 . . PUNCT cana-1326 213 1 thus thus ADV cana-1326 213 2 we we PRON cana-1326 213 3 obtain obtain VERB cana-1326 213 4 that that SCONJ cana-1326 213 5 k1 k1 NOUN cana-1326 213 6 οΏ½ οΏ½ PROPN cana-1326 213 7 = = SYM cana-1326 213 8 sl1 sl1 PROPN cana-1326 213 9 = = PROPN cana-1326 213 10 sk1 sk1 PROPN cana-1326 213 11 οΏ½ οΏ½ PROPN cana-1326 213 12 . . PUNCT cana-1326 214 1 similarly similarly ADV cana-1326 214 2 , , PUNCT cana-1326 214 3 one one PRON cana-1326 214 4 can can AUX cana-1326 214 5 also also ADV cana-1326 214 6 have have VERB cana-1326 214 7 tk2 tk2 NOUN cana-1326 214 8 = = SYM cana-1326 214 9 k2 k2 PROPN cana-1326 214 10 . . PUNCT cana-1326 215 1 as as SCONJ cana-1326 215 2 k1 k1 PROPN cana-1326 215 3 = = SYM cana-1326 215 4 k2 k2 PROPN cana-1326 215 5 implies imply VERB cana-1326 215 6 sk1 sk1 PROPN cana-1326 215 7 = = SYM cana-1326 215 8 tk1 tk1 PROPN cana-1326 215 9 = = SYM cana-1326 215 10 k1 k1 PROPN cana-1326 215 11 , , PUNCT cana-1326 215 12 therefore therefore ADV cana-1326 215 13 k1 k1 PROPN cana-1326 215 14 = = SYM cana-1326 215 15 k2 k2 PROPN cana-1326 215 16 is be AUX cana-1326 215 17 common common ADJ cana-1326 215 18 fixed fix VERB cana-1326 215 19 point point NOUN cana-1326 215 20 of of ADP cana-1326 215 21 s s PRON cana-1326 215 22 and and CCONJ cana-1326 215 23 t. t. PROPN cana-1326 215 24 for for ADP cana-1326 215 25 uniqueness uniqueness NOUN cana-1326 215 26 assume assume VERB cana-1326 215 27 that that SCONJ cana-1326 215 28 k1 k1 NOUN cana-1326 215 29 βˆ— βˆ— NOUN cana-1326 215 30 in in ADP cana-1326 215 31 x x PRON cana-1326 215 32 is be AUX cana-1326 215 33 another another DET cana-1326 215 34 common common ADJ cana-1326 215 35 fixed fix VERB cana-1326 215 36 point point NOUN cana-1326 215 37 of of ADP cana-1326 215 38 s s PRON cana-1326 215 39 and and CCONJ cana-1326 215 40 t. t. NOUN cana-1326 215 41 then then ADV cana-1326 215 42 we we PRON cana-1326 215 43 have have VERB cana-1326 215 44 sk1 sk1 PROPN cana-1326 215 45 βˆ—=tk1 βˆ—=tk1 PROPN cana-1326 215 46 βˆ— βˆ— NOUN cana-1326 215 47 = = SYM cana-1326 215 48 k1 k1 NOUN cana-1326 215 49 βˆ— βˆ— NOUN cana-1326 215 50 as as ADP cana-1326 215 51 𝑑 𝑑 PROPN cana-1326 215 52 οΏ½ οΏ½ NOUN cana-1326 215 53 (k1 (k1 SYM cana-1326 215 54 , , PUNCT cana-1326 215 55 , , PUNCT cana-1326 215 56 𝑇 𝑇 PROPN cana-1326 215 57 οΏ½ οΏ½ PROPN cana-1326 215 58 k1 k1 NOUN cana-1326 215 59 βˆ— βˆ— NOUN cana-1326 215 60 ) ) PUNCT cana-1326 216 1 + + CCONJ cana-1326 216 2 𝑑 𝑑 PROPN cana-1326 216 3 οΏ½ οΏ½ PROPN cana-1326 216 4 (k1 (k1 NOUN cana-1326 216 5 βˆ— βˆ— NOUN cana-1326 216 6 , , PUNCT cana-1326 216 7 𝑆 𝑆 PROPN cana-1326 216 8 οΏ½ οΏ½ PROPN cana-1326 216 9 k1 k1 PROPN cana-1326 216 10 , , PUNCT cana-1326 216 11 ) ) PUNCT cana-1326 217 1 + + CCONJ cana-1326 217 2 𝑑 𝑑 PROPN cana-1326 217 3 οΏ½ οΏ½ NOUN cana-1326 217 4 (k1 (k1 SYM cana-1326 217 5 , , PUNCT cana-1326 217 6 , , PUNCT cana-1326 217 7 k1 k1 PROPN cana-1326 217 8 βˆ—)= βˆ—)= PROPN cana-1326 217 9 0 0 NUM cana-1326 217 10 , , PUNCT cana-1326 217 11 therefore therefore ADV cana-1326 217 12 𝑑 𝑑 PROPN cana-1326 217 13 οΏ½ οΏ½ NOUN cana-1326 217 14 (sk1 (sk1 NOUN cana-1326 217 15 , , PUNCT cana-1326 217 16 tk1 tk1 PROPN cana-1326 217 17 βˆ— βˆ— NOUN cana-1326 217 18 ) ) PUNCT cana-1326 218 1 = = PUNCT cana-1326 218 2 𝑑 𝑑 PROPN cana-1326 218 3 οΏ½ οΏ½ PROPN cana-1326 218 4 (k1 (k1 SYM cana-1326 218 5 , , PUNCT cana-1326 218 6 , , PUNCT cana-1326 218 7 k1 k1 PROPN cana-1326 218 8 βˆ—)=0 βˆ—)=0 NOUN cana-1326 218 9 . . PUNCT cana-1326 219 1 this this PRON cana-1326 219 2 implies imply VERB cana-1326 219 3 that that SCONJ cana-1326 219 4 k1 k1 NOUN cana-1326 219 5 , , PUNCT cana-1326 219 6 = = SYM cana-1326 219 7 k1 k1 NOUN cana-1326 219 8 βˆ—. βˆ—. PROPN cana-1326 219 9 this this PRON cana-1326 219 10 completes complete VERB cana-1326 219 11 the the DET cana-1326 219 12 proof proof NOUN cana-1326 219 13 of of ADP cana-1326 219 14 the the DET cana-1326 219 15 theorem theorem NOUN cana-1326 219 16 . . PUNCT cana-1326 220 1 corollary corollary ADJ cana-1326 220 2 3.1 3.1 NUM cana-1326 220 3 : : PUNCT cana-1326 220 4 let let VERB cana-1326 220 5 ( ( PUNCT cana-1326 220 6 𝑋 𝑋 PROPN cana-1326 220 7 οΏ½ οΏ½ PROPN cana-1326 220 8 , , PUNCT cana-1326 220 9 𝑑 𝑑 NOUN cana-1326 220 10 οΏ½ οΏ½ NOUN cana-1326 220 11 ) ) PUNCT cana-1326 220 12 be be VERB cana-1326 220 13 a a DET cana-1326 220 14 complete complete ADJ cana-1326 220 15 bicomplex bicomplex NOUN cana-1326 220 16 valued value VERB cana-1326 220 17 𝑏 𝑏 DET cana-1326 220 18 οΏ½ οΏ½ NOUN cana-1326 220 19 -metric -metric ADJ cana-1326 220 20 space space NOUN cana-1326 220 21 with with ADP cana-1326 220 22 the the DET cana-1326 220 23 coefficient coefficient NOUN cana-1326 220 24 𝑠 𝑠 PROPN cana-1326 220 25 οΏ½ οΏ½ PROPN cana-1326 220 26 β‰₯1 β‰₯1 PROPN cana-1326 220 27 and and CCONJ cana-1326 220 28 let let VERB cana-1326 220 29 t t PROPN cana-1326 220 30 : : PUNCT cana-1326 220 31 𝑋 𝑋 PROPN cana-1326 220 32 οΏ½ οΏ½ PROPN cana-1326 220 33 β†’ β†’ SYM cana-1326 220 34 𝑋 𝑋 PROPN cana-1326 220 35 οΏ½ οΏ½ PROPN cana-1326 220 36 be be AUX cana-1326 220 37 mappings mapping NOUN cana-1326 220 38 satisfying satisfy VERB cana-1326 220 39 d(tx d(tx PROPN cana-1326 220 40 , , PUNCT cana-1326 220 41 ty ty NOUN cana-1326 220 42 ) ) PUNCT cana-1326 220 43 ≲𝑖2ad(x ≲𝑖2ad(x PUNCT cana-1326 220 44 , , PUNCT cana-1326 220 45 y)+b y)+b PROPN cana-1326 220 46 𝑑(π‘₯,𝑇π‘₯)𝑑(𝑦,𝑇𝑦 𝑑(π‘₯,𝑇π‘₯)𝑑(𝑦,𝑇𝑦 NOUN cana-1326 220 47 ) ) PUNCT cana-1326 221 1 d d X cana-1326 221 2 οΏ½ οΏ½ PROPN cana-1326 221 3 (x (x PROPN cana-1326 221 4 , , PUNCT cana-1326 221 5 ty) ty) PROPN cana-1326 221 6 οΏ½ οΏ½ PROPN cana-1326 221 7 + + NOUN cana-1326 221 8 οΏ½ οΏ½ PROPN cana-1326 221 9 d d NOUN cana-1326 221 10 οΏ½ οΏ½ PROPN cana-1326 221 11 (y (y NOUN cana-1326 221 12 , , PUNCT cana-1326 221 13 tx) tx) PROPN cana-1326 221 14 οΏ½ οΏ½ PROPN cana-1326 221 15 + + PROPN cana-1326 221 16 οΏ½ οΏ½ PROPN cana-1326 221 17 d d NOUN cana-1326 221 18 οΏ½ οΏ½ PROPN cana-1326 221 19 (x (x PROPN cana-1326 221 20 , , PUNCT cana-1326 221 21 y y NOUN cana-1326 221 22 ) ) PUNCT cana-1326 221 23 for for ADP cana-1326 221 24 all all DET cana-1326 221 25 π‘₯ π‘₯ PROPN cana-1326 221 26 οΏ½ οΏ½ PROPN cana-1326 221 27 , , PUNCT cana-1326 221 28 𝑦 𝑦 NOUN cana-1326 221 29 οΏ½ οΏ½ PROPN cana-1326 221 30 ∈ ∈ PROPN cana-1326 221 31 𝑋 𝑋 PROPN cana-1326 221 32 οΏ½ οΏ½ PROPN cana-1326 221 33 , , PUNCT cana-1326 221 34 such such ADJ cana-1326 221 35 that that SCONJ cana-1326 221 36 xβ‰  xβ‰  PROPN cana-1326 221 37 𝑦 𝑦 PROPN cana-1326 221 38 , , PUNCT cana-1326 221 39 𝑑 𝑑 PROPN cana-1326 221 40 οΏ½ οΏ½ PROPN cana-1326 221 41 (π‘₯ (π‘₯ PROPN cana-1326 221 42 οΏ½ οΏ½ PROPN cana-1326 221 43 , , PUNCT cana-1326 221 44 𝑇 𝑇 PROPN cana-1326 221 45 οΏ½ οΏ½ PROPN cana-1326 221 46 𝑦 𝑦 NOUN cana-1326 221 47 οΏ½ οΏ½ PROPN cana-1326 221 48 )+𝑑 )+𝑑 PROPN cana-1326 221 49 οΏ½ οΏ½ PROPN cana-1326 221 50 (𝑦 (𝑦 PROPN cana-1326 221 51 οΏ½ οΏ½ PROPN cana-1326 221 52 , , PUNCT cana-1326 221 53 tπ‘₯ tπ‘₯ PROPN cana-1326 221 54 οΏ½ οΏ½ PROPN cana-1326 221 55 )+𝑑 )+𝑑 PROPN cana-1326 221 56 οΏ½ οΏ½ PROPN cana-1326 221 57 (π‘₯ (π‘₯ PROPN cana-1326 221 58 οΏ½ οΏ½ PROPN cana-1326 221 59 , , PUNCT cana-1326 221 60 𝑦 𝑦 NOUN cana-1326 221 61 οΏ½ οΏ½ NOUN cana-1326 221 62 ) ) PUNCT cana-1326 221 63 β‰  β‰  PROPN cana-1326 221 64 0 0 NUM cana-1326 221 65 , , PUNCT cana-1326 221 66 where where SCONJ cana-1326 221 67 a a DET cana-1326 221 68 , , PUNCT cana-1326 221 69 b b NOUN cana-1326 221 70 are be AUX cana-1326 221 71 nonnegative nonnegative ADJ cana-1326 221 72 reals real NOUN cana-1326 221 73 with with ADP cana-1326 221 74 a+ a+ PRON cana-1326 221 75 √2𝑠 √2𝑠 NUM cana-1326 221 76 οΏ½ οΏ½ NOUN cana-1326 221 77 b b NOUN cana-1326 221 78 < < X cana-1326 221 79 1 1 NUM cana-1326 221 80 or or CCONJ cana-1326 221 81 𝑑 𝑑 NOUN cana-1326 221 82 οΏ½ οΏ½ PROPN cana-1326 221 83 (tπ‘₯ (tπ‘₯ SYM cana-1326 221 84 οΏ½ οΏ½ PROPN cana-1326 221 85 , , PUNCT cana-1326 221 86 𝑇 𝑇 PROPN cana-1326 221 87 οΏ½ οΏ½ PROPN cana-1326 221 88 𝑦 𝑦 NOUN cana-1326 221 89 οΏ½ οΏ½ PROPN cana-1326 221 90 ) ) PUNCT cana-1326 221 91 = = SYM cana-1326 221 92 0 0 PUNCT cana-1326 222 1 if if SCONJ cana-1326 222 2 𝑑 𝑑 PROPN cana-1326 222 3 οΏ½ οΏ½ PROPN cana-1326 222 4 (π‘₯ (π‘₯ PROPN cana-1326 222 5 οΏ½ οΏ½ PROPN cana-1326 222 6 , , PUNCT cana-1326 222 7 𝑇 𝑇 PROPN cana-1326 222 8 οΏ½ οΏ½ PROPN cana-1326 222 9 𝑦 𝑦 NOUN cana-1326 222 10 οΏ½ οΏ½ PROPN cana-1326 222 11 ) ) PUNCT cana-1326 222 12 + + CCONJ cana-1326 222 13 𝑑 𝑑 PROPN cana-1326 222 14 οΏ½ οΏ½ PROPN cana-1326 222 15 (𝑦 (𝑦 NOUN cana-1326 222 16 οΏ½ οΏ½ PROPN cana-1326 222 17 , , PUNCT cana-1326 222 18 tπ‘₯ tπ‘₯ PROPN cana-1326 222 19 οΏ½ οΏ½ PROPN cana-1326 222 20 ) ) PUNCT cana-1326 222 21 + + CCONJ cana-1326 222 22 𝑑 𝑑 PROPN cana-1326 222 23 οΏ½ οΏ½ PROPN cana-1326 222 24 (π‘₯ (π‘₯ PROPN cana-1326 222 25 οΏ½ οΏ½ PROPN cana-1326 222 26 , , PUNCT cana-1326 222 27 𝑦 𝑦 NOUN cana-1326 222 28 οΏ½ οΏ½ NOUN cana-1326 222 29 ) ) PUNCT cana-1326 222 30 = = SYM cana-1326 223 1 0 0 X cana-1326 223 2 . . PUNCT cana-1326 223 3 then then ADV cana-1326 223 4 has have VERB cana-1326 223 5 a a DET cana-1326 223 6 unique unique ADJ cana-1326 223 7 common common ADJ cana-1326 223 8 fixed fix VERB cana-1326 223 9 point point NOUN cana-1326 223 10 . . PUNCT cana-1326 224 1 proof proof NOUN cana-1326 224 2 : : PUNCT cana-1326 224 3 we we PRON cana-1326 224 4 can can AUX cana-1326 224 5 prove prove VERB cana-1326 224 6 this this DET cana-1326 224 7 result result NOUN cana-1326 224 8 by by ADP cana-1326 224 9 applying apply VERB cana-1326 224 10 theorem theorem NOUN cana-1326 224 11 3 3 NUM cana-1326 224 12 by by ADP cana-1326 224 13 setting set VERB cana-1326 224 14 s s PART cana-1326 224 15 = = ADJ cana-1326 224 16 t. t. NOUN cana-1326 224 17 corollary corollary NOUN cana-1326 224 18 3.2 3.2 NUM cana-1326 224 19 : : PUNCT cana-1326 224 20 let let VERB cana-1326 224 21 ( ( PUNCT cana-1326 224 22 𝑋 𝑋 PROPN cana-1326 224 23 οΏ½ οΏ½ PROPN cana-1326 224 24 , , PUNCT cana-1326 224 25 𝑑 𝑑 NOUN cana-1326 224 26 οΏ½ οΏ½ NOUN cana-1326 224 27 ) ) PUNCT cana-1326 224 28 be be VERB cana-1326 224 29 a a DET cana-1326 224 30 complete complete ADJ cana-1326 224 31 bicomplex bicomplex NOUN cana-1326 224 32 valued value VERB cana-1326 224 33 𝑏 𝑏 DET cana-1326 224 34 οΏ½ οΏ½ NOUN cana-1326 224 35 -metric -metric ADJ cana-1326 224 36 space space NOUN cana-1326 224 37 with with ADP cana-1326 224 38 the the DET cana-1326 224 39 coefficient coefficient NOUN cana-1326 224 40 𝑠 𝑠 PROPN cana-1326 224 41 οΏ½ οΏ½ PROPN cana-1326 224 42 β‰₯1 β‰₯1 PROPN cana-1326 224 43 and and CCONJ cana-1326 224 44 let let VERB cana-1326 224 45 t t PROPN cana-1326 224 46 : : PUNCT cana-1326 224 47 𝑋 𝑋 PROPN cana-1326 224 48 οΏ½ οΏ½ PROPN cana-1326 224 49 β†’ β†’ SYM cana-1326 224 50 𝑋 𝑋 PROPN cana-1326 224 51 οΏ½ οΏ½ PROPN cana-1326 224 52 be be AUX cana-1326 224 53 mappings mapping NOUN cana-1326 224 54 satisfying satisfy VERB cana-1326 224 55 ( ( PUNCT cana-1326 224 56 for for ADP cana-1326 224 57 some some DET cana-1326 224 58 fixed fix VERB cana-1326 224 59 n n CCONJ cana-1326 224 60 ) ) PUNCT cana-1326 224 61 d(𝑇𝑛x, d(𝑇𝑛x, NOUN cana-1326 224 62 οΏ½ οΏ½ NOUN cana-1326 224 63 𝑇𝑛y 𝑇𝑛y ADJ cana-1326 224 64 ) ) PUNCT cana-1326 224 65 ≲𝑖2ad(x ≲𝑖2ad(x PUNCT cana-1326 224 66 , , PUNCT cana-1326 225 1 y)+b y)+b PROPN cana-1326 225 2 𝑑(π‘₯,𝑇𝑛π‘₯)𝑑(𝑦,𝑇𝑛𝑦 𝑑(π‘₯,𝑇𝑛π‘₯)𝑑(𝑦,𝑇𝑛𝑦 NOUN cana-1326 225 3 ) ) PUNCT cana-1326 225 4 d d NOUN cana-1326 225 5 οΏ½ οΏ½ PROPN cana-1326 225 6 (x,𝑇𝑛y) (x,𝑇𝑛y) PROPN cana-1326 225 7 οΏ½ οΏ½ PROPN cana-1326 225 8 + + PROPN cana-1326 225 9 οΏ½ οΏ½ PROPN cana-1326 225 10 d d NOUN cana-1326 225 11 οΏ½ οΏ½ PROPN cana-1326 225 12 (y,𝑇𝑛x) (y,𝑇𝑛x) NOUN cana-1326 225 13 οΏ½ οΏ½ PROPN cana-1326 225 14 + + PROPN cana-1326 225 15 οΏ½ οΏ½ PROPN cana-1326 225 16 d d NOUN cana-1326 225 17 οΏ½ οΏ½ PROPN cana-1326 225 18 (x (x PROPN cana-1326 225 19 , , PUNCT cana-1326 225 20 y y NOUN cana-1326 225 21 ) ) PUNCT cana-1326 225 22 for for ADP cana-1326 225 23 all all DET cana-1326 225 24 π‘₯ π‘₯ PROPN cana-1326 225 25 οΏ½ οΏ½ PROPN cana-1326 225 26 , , PUNCT cana-1326 225 27 𝑦 𝑦 NOUN cana-1326 225 28 οΏ½ οΏ½ PROPN cana-1326 225 29 ∈ ∈ PROPN cana-1326 225 30 𝑋 𝑋 PROPN cana-1326 225 31 οΏ½ οΏ½ PROPN cana-1326 225 32 , , PUNCT cana-1326 226 1 such such ADJ cana-1326 226 2 that that SCONJ cana-1326 226 3 xβ‰  xβ‰  PROPN cana-1326 226 4 𝑦 𝑦 PROPN cana-1326 226 5 , , PUNCT cana-1326 226 6 𝑑 𝑑 PROPN cana-1326 226 7 οΏ½ οΏ½ PROPN cana-1326 226 8 (π‘₯ (π‘₯ PROPN cana-1326 226 9 οΏ½ οΏ½ PROPN cana-1326 226 10 , , PUNCT cana-1326 226 11 𝑇𝑛𝑦 𝑇𝑛𝑦 PROPN cana-1326 226 12 οΏ½ οΏ½ PROPN cana-1326 226 13 )+𝑑 )+𝑑 PROPN cana-1326 226 14 οΏ½ οΏ½ PROPN cana-1326 226 15 (𝑦 (𝑦 PROPN cana-1326 226 16 οΏ½ οΏ½ PROPN cana-1326 226 17 , , PUNCT cana-1326 226 18 𝑇𝑛π‘₯ 𝑇𝑛π‘₯ PROPN cana-1326 226 19 οΏ½ οΏ½ PROPN cana-1326 226 20 )+𝑑 )+𝑑 PROPN cana-1326 226 21 οΏ½ οΏ½ PROPN cana-1326 226 22 (π‘₯ (π‘₯ PROPN cana-1326 226 23 οΏ½ οΏ½ PROPN cana-1326 226 24 , , PUNCT cana-1326 226 25 𝑦 𝑦 NOUN cana-1326 226 26 οΏ½ οΏ½ NOUN cana-1326 226 27 ) ) PUNCT cana-1326 226 28 β‰  β‰  PROPN cana-1326 226 29 0 0 NUM cana-1326 226 30 , , PUNCT cana-1326 226 31 where where SCONJ cana-1326 226 32 a a DET cana-1326 226 33 , , PUNCT cana-1326 226 34 b b NOUN cana-1326 226 35 are be AUX cana-1326 226 36 nonnegative nonnegative ADJ cana-1326 226 37 reals real NOUN cana-1326 226 38 with with ADP cana-1326 226 39 a+ a+ DET cana-1326 226 40 √2𝑠 √2𝑠 NUM cana-1326 226 41 οΏ½ οΏ½ NOUN cana-1326 226 42 b b NOUN cana-1326 226 43 < < X cana-1326 226 44 1 1 NUM cana-1326 226 45 or or CCONJ cana-1326 226 46 𝑑 𝑑 NOUN cana-1326 226 47 οΏ½ οΏ½ PROPN cana-1326 226 48 (𝑇𝑛π‘₯ (𝑇𝑛π‘₯ SYM cana-1326 226 49 οΏ½ οΏ½ PROPN cana-1326 226 50 , , PUNCT cana-1326 226 51 𝑇𝑛𝑦 𝑇𝑛𝑦 NOUN cana-1326 226 52 οΏ½ οΏ½ NOUN cana-1326 226 53 ) ) PUNCT cana-1326 226 54 = = SYM cana-1326 226 55 0 0 PUNCT cana-1326 227 1 if if SCONJ cana-1326 227 2 𝑑 𝑑 PROPN cana-1326 227 3 οΏ½ οΏ½ PROPN cana-1326 227 4 (π‘₯ (π‘₯ PROPN cana-1326 227 5 οΏ½ οΏ½ PROPN cana-1326 227 6 , , PUNCT cana-1326 227 7 𝑇𝑛𝑦 𝑇𝑛𝑦 PROPN cana-1326 227 8 οΏ½ οΏ½ PROPN cana-1326 227 9 ) ) PUNCT cana-1326 227 10 + + CCONJ cana-1326 227 11 𝑑 𝑑 PROPN cana-1326 227 12 οΏ½ οΏ½ PROPN cana-1326 227 13 (𝑦 (𝑦 NOUN cana-1326 227 14 οΏ½ οΏ½ PROPN cana-1326 227 15 , , PUNCT cana-1326 227 16 𝑇𝑛π‘₯ 𝑇𝑛π‘₯ PROPN cana-1326 227 17 οΏ½ οΏ½ PROPN cana-1326 227 18 ) ) PUNCT cana-1326 227 19 + + CCONJ cana-1326 227 20 𝑑 𝑑 PROPN cana-1326 227 21 οΏ½ οΏ½ PROPN cana-1326 227 22 (π‘₯ (π‘₯ PROPN cana-1326 227 23 οΏ½ οΏ½ PROPN cana-1326 227 24 , , PUNCT cana-1326 227 25 𝑦 𝑦 NOUN cana-1326 227 26 οΏ½ οΏ½ NOUN cana-1326 227 27 ) ) PUNCT cana-1326 227 28 = = SYM cana-1326 228 1 0 0 X cana-1326 228 2 . . PUNCT cana-1326 228 3 then then ADV cana-1326 228 4 has have VERB cana-1326 228 5 a a DET cana-1326 228 6 unique unique ADJ cana-1326 228 7 common common ADJ cana-1326 228 8 fixed fix VERB cana-1326 228 9 point point NOUN cana-1326 228 10 . . PUNCT cana-1326 229 1 references reference NOUN cana-1326 229 2 : : PUNCT cana-1326 230 1 [ [ X cana-1326 230 2 1 1 X cana-1326 230 3 ] ] PUNCT cana-1326 230 4 j. j. PROPN cana-1326 230 5 choi choi PROPN cana-1326 230 6 _ _ PROPN cana-1326 230 7 , , PUNCT cana-1326 230 8 s. s. PROPN cana-1326 230 9 k. k. PROPN cana-1326 230 10 datta datta PROPN cana-1326 230 11 , , PUNCT cana-1326 230 12 t. t. PROPN cana-1326 230 13 biswas biswas PROPN cana-1326 230 14 and and CCONJ cana-1326 230 15 md md PROPN cana-1326 230 16 n. n. PROPN cana-1326 230 17 islam islam PROPN cana-1326 230 18 : : PUNCT cana-1326 230 19 some some DET cana-1326 230 20 fixed fix VERB cana-1326 230 21 point point NOUN cana-1326 230 22 theorems theorem NOUN cana-1326 230 23 in in ADP cana-1326 230 24 connectionwith connectionwith NOUN cana-1326 230 25 two two NUM cana-1326 230 26 weakly weakly ADJ cana-1326 230 27 compatible compatible ADJ cana-1326 230 28 mappings mapping NOUN cana-1326 230 29 in in ADP cana-1326 230 30 bicomplex bicomplex NOUN cana-1326 230 31 valued value VERB cana-1326 230 32 metric metric ADJ cana-1326 230 33 spaces space NOUN cana-1326 230 34 , , PUNCT cana-1326 230 35 honam honam PROPN cana-1326 230 36 mathematical mathematical PROPN cana-1326 230 37 j. j. PROPN cana-1326 230 38 39 39 NUM cana-1326 230 39 ( ( PUNCT cana-1326 230 40 2017 2017 NUM cana-1326 230 41 ) ) PUNCT cana-1326 230 42 , , PUNCT cana-1326 230 43 no no INTJ cana-1326 230 44 . . NOUN cana-1326 230 45 1 1 NUM cana-1326 230 46 , , PUNCT cana-1326 230 47 pp pp ADJ cana-1326 230 48 . . PUNCT cana-1326 231 1 115 115 NUM cana-1326 231 2 - - SYM cana-1326 231 3 126 126 NUM cana-1326 231 4 . . PUNCT cana-1326 232 1 [ [ X cana-1326 232 2 2 2 NUM cana-1326 232 3 ] ] PUNCT cana-1326 232 4 a. a. NOUN cana-1326 232 5 azam azam PROPN cana-1326 232 6 , , PUNCT cana-1326 232 7 f. f. PROPN cana-1326 232 8 brain brain PROPN cana-1326 232 9 and and CCONJ cana-1326 232 10 m. m. PROPN cana-1326 232 11 khan khan PROPN cana-1326 232 12 : : PUNCT cana-1326 232 13 common common ADJ cana-1326 232 14 fixed fix VERB cana-1326 232 15 point point NOUN cana-1326 232 16 theorems theorem NOUN cana-1326 232 17 in in ADP cana-1326 232 18 complex complex ADJ cana-1326 232 19 valued value VERB cana-1326 232 20 metric metric ADJ cana-1326 232 21 spaces space NOUN cana-1326 232 22 , , PUNCT cana-1326 232 23 numer numer PROPN cana-1326 232 24 . . PUNCT cana-1326 233 1 funct funct PROPN cana-1326 233 2 . . PUNCT cana-1326 234 1 anal anal PROPN cana-1326 234 2 . . PUNCT cana-1326 235 1 optim optim PROPN cana-1326 235 2 . . PUNCT cana-1326 236 1 32 32 NUM cana-1326 236 2 ( ( PUNCT cana-1326 236 3 3 3 NUM cana-1326 236 4 ) ) PUNCT cana-1326 236 5 ( ( PUNCT cana-1326 236 6 2011 2011 NUM cana-1326 236 7 ) ) PUNCT cana-1326 236 8 , , PUNCT cana-1326 236 9 243 243 NUM cana-1326 236 10 - - SYM cana-1326 236 11 253 253 NUM cana-1326 236 12 . . PUNCT cana-1326 237 1 [ [ X cana-1326 237 2 3 3 X cana-1326 237 3 ] ] X cana-1326 237 4 c. c. NOUN cana-1326 237 5 segre segre NOUN cana-1326 237 6 , , PUNCT cana-1326 237 7 le le X cana-1326 237 8 rappresentazioni rappresentazioni PROPN cana-1326 237 9 reali reali PROPN cana-1326 237 10 delle delle PROPN cana-1326 237 11 forme forme PROPN cana-1326 237 12 complesse complesse PROPN cana-1326 237 13 e e PROPN cana-1326 237 14 gli gli NOUN cana-1326 237 15 enti enti X cana-1326 237 16 iperalgebrici iperalgebrici NOUN cana-1326 237 17 , , PUNCT cana-1326 237 18 math math NOUN cana-1326 237 19 . . PUNCT cana-1326 238 1 ann ann PROPN cana-1326 238 2 . . PROPN cana-1326 239 1 40 40 NUM cana-1326 239 2 ( ( PUNCT cana-1326 239 3 1892 1892 NUM cana-1326 239 4 ) ) PUNCT cana-1326 239 5 , , PUNCT cana-1326 239 6 413 413 NUM cana-1326 239 7 - - SYM cana-1326 239 8 467 467 NUM cana-1326 239 9 . . PUNCT cana-1326 240 1 [ [ X cana-1326 240 2 4 4 NUM cana-1326 240 3 ] ] X cana-1326 240 4 n. n. PROPN cana-1326 240 5 spampinato spampinato PROPN cana-1326 240 6 , , PUNCT cana-1326 240 7 estensione estensione PROPN cana-1326 240 8 nel nel PROPN cana-1326 240 9 campo campo PROPN cana-1326 240 10 bicomplesso bicomplesso PROPN cana-1326 240 11 di di PROPN cana-1326 240 12 due due ADJ cana-1326 240 13 teoremi teoremi NOUN cana-1326 240 14 , , PUNCT cana-1326 240 15 del del PROPN cana-1326 240 16 levi levi PROPN cana-1326 240 17 - - PUNCT cana-1326 240 18 civita civita PROPN cana-1326 240 19 e e PROPN cana-1326 240 20 del del PROPN cana-1326 240 21 severi severi PROPN cana-1326 240 22 , , PUNCT cana-1326 240 23 per per X cana-1326 240 24 le le X cana-1326 240 25 funzioni funzioni PROPN cana-1326 240 26 olomorfe olomorfe PROPN cana-1326 240 27 di di PROPN cana-1326 240 28 due due ADJ cana-1326 240 29 variablili variablili NOUN cana-1326 240 30 bicomplesse bicomplesse NOUN cana-1326 240 31 i i PROPN cana-1326 240 32 , , PUNCT cana-1326 240 33 ii ii PROPN cana-1326 240 34 , , PUNCT cana-1326 240 35 reale reale PROPN cana-1326 240 36 accad accad PROPN cana-1326 240 37 . . PUNCT cana-1326 241 1 naz naz PROPN cana-1326 241 2 . . PUNCT cana-1326 242 1 lincei lincei NOUN cana-1326 242 2 22(6 22(6 PROPN cana-1326 242 3 ) ) PUNCT cana-1326 242 4 ( ( PUNCT cana-1326 242 5 1935 1935 NUM cana-1326 242 6 ) ) PUNCT cana-1326 242 7 , , PUNCT cana-1326 242 8 38 38 NUM cana-1326 242 9 - - SYM cana-1326 242 10 43 43 NUM cana-1326 242 11 , , PUNCT cana-1326 242 12 96 96 NUM cana-1326 242 13 - - SYM cana-1326 242 14 102 102 NUM cana-1326 242 15 . . PUNCT cana-1326 243 1 [ [ X cana-1326 243 2 5 5 NUM cana-1326 243 3 ] ] X cana-1326 243 4 n. n. PROPN cana-1326 243 5 spampinato spampinato PROPN cana-1326 243 6 , , PUNCT cana-1326 243 7 sulla sulla PROPN cana-1326 243 8 rappresentazione rappresentazione PROPN cana-1326 243 9 delle delle PROPN cana-1326 243 10 funzioni funzioni PROPN cana-1326 243 11 do do AUX cana-1326 243 12 variabile variabile NOUN cana-1326 243 13 bicomplessa bicomplessa NOUN cana-1326 243 14 totalmente totalmente PROPN cana-1326 243 15 derivabili derivabili PROPN cana-1326 243 16 , , PUNCT cana-1326 243 17 ann ann PROPN cana-1326 243 18 . . PROPN cana-1326 243 19 mat mat PROPN cana-1326 243 20 . . PUNCT cana-1326 243 21 pura pura NOUN cana-1326 243 22 appl appl NOUN cana-1326 243 23 . . PUNCT cana-1326 243 24 14(4 14(4 NUM cana-1326 243 25 ) ) PUNCT cana-1326 243 26 ( ( PUNCT cana-1326 243 27 1936 1936 NUM cana-1326 243 28 ) ) PUNCT cana-1326 243 29 , , PUNCT cana-1326 243 30 305 305 NUM cana-1326 243 31 - - SYM cana-1326 243 32 325 325 NUM cana-1326 243 33 . . PUNCT cana-1326 244 1 [ [ X cana-1326 244 2 6 6 NUM cana-1326 244 3 ] ] PUNCT cana-1326 244 4 i. i. NOUN cana-1326 244 5 beg beg PROPN cana-1326 244 6 , , PUNCT cana-1326 244 7 s.k s.k PROPN cana-1326 244 8 . . PROPN cana-1326 244 9 datta datta PROPN cana-1326 244 10 and and CCONJ cana-1326 244 11 d. d. PROPN cana-1326 244 12 pal pal PROPN cana-1326 244 13 : : PUNCT cana-1326 244 14 fixed fix VERB cana-1326 244 15 point point NOUN cana-1326 244 16 in in ADP cana-1326 244 17 bicomplex bicomplex NOUN cana-1326 244 18 valued value VERB cana-1326 244 19 metric metric ADJ cana-1326 244 20 spaces space NOUN cana-1326 244 21 , , PUNCT cana-1326 244 22 int int NOUN cana-1326 244 23 . . PUNCT cana-1326 245 1 j. j. PROPN cana-1326 245 2 nonlinear nonlinear PROPN cana-1326 245 3 anal anal PROPN cana-1326 245 4 . . PUNCT cana-1326 246 1 appl appl PROPN cana-1326 246 2 . . PUNCT cana-1326 247 1 12(2021 12(2021 NUM cana-1326 247 2 ) ) PUNCT cana-1326 247 3 , , PUNCT cana-1326 247 4 no.2 no.2 PROPN cana-1326 247 5 , , PUNCT cana-1326 247 6 717 717 NUM cana-1326 247 7 - - SYM cana-1326 247 8 727 727 NUM cana-1326 247 9 . . PUNCT cana-1326 248 1 [ [ X cana-1326 248 2 7 7 NUM cana-1326 248 3 ] ] X cana-1326 248 4 a. a. NOUN cana-1326 248 5 singh singh PROPN cana-1326 248 6 , , PUNCT cana-1326 248 7 m.s.khan m.s.khan PROPN cana-1326 248 8 and and CCONJ cana-1326 248 9 b. b. PROPN cana-1326 248 10 fisher fisher PROPN cana-1326 248 11 : : PUNCT cana-1326 248 12 some some DET cana-1326 248 13 fixed fix VERB cana-1326 248 14 point point NOUN cana-1326 248 15 theorems theorem NOUN cana-1326 248 16 for for ADP cana-1326 248 17 certain certain ADJ cana-1326 248 18 contractive contractive ADJ cana-1326 248 19 mapping mapping NOUN cana-1326 248 20 on on ADP cana-1326 248 21 metric metric ADJ cana-1326 248 22 and and CCONJ cana-1326 248 23 generalized generalized ADJ cana-1326 248 24 metric metric ADJ cana-1326 248 25 space space NOUN cana-1326 248 26 , , PUNCT cana-1326 248 27 mathematical mathematical ADJ cana-1326 248 28 moravia,16 moravia,16 PROPN cana-1326 248 29 - - PUNCT cana-1326 248 30 2(2012),69 2(2012),69 NUM cana-1326 248 31 - - SYM cana-1326 248 32 77 77 NUM cana-1326 248 33 . . PUNCT cana-1326 249 1 communications communication NOUN cana-1326 249 2 on on ADP cana-1326 249 3 applied apply VERB cana-1326 249 4 nonlinear nonlinear ADJ cana-1326 249 5 analysis analysis NOUN cana-1326 249 6 issn issn NOUN cana-1326 249 7 : : PUNCT cana-1326 249 8 1074 1074 NUM cana-1326 249 9 - - PUNCT cana-1326 249 10 133x 133x NUM cana-1326 249 11 vol vol NOUN cana-1326 249 12 31 31 NUM cana-1326 249 13 no no NOUN cana-1326 249 14 . . PUNCT cana-1326 250 1 7s 7 NOUN cana-1326 250 2 ( ( PUNCT cana-1326 250 3 2024 2024 NUM cana-1326 250 4 ) ) PUNCT cana-1326 250 5 477 477 NUM cana-1326 250 6 https://internationalpubls.com https://internationalpubls.com X cana-1326 251 1 [ [ X cana-1326 251 2 8 8 NUM cana-1326 251 3 ] ] X cana-1326 251 4 i.h i.h PROPN cana-1326 251 5 . . PROPN cana-1326 251 6 jebril jebril NOUN cana-1326 251 7 , , PUNCT cana-1326 251 8 s.k s.k PROPN cana-1326 251 9 . . PROPN cana-1326 251 10 datta datta PROPN cana-1326 251 11 , , PUNCT cana-1326 251 12 r. r. PROPN cana-1326 251 13 sarkar sarkar PROPN cana-1326 251 14 and and CCONJ cana-1326 251 15 n. n. PROPN cana-1326 251 16 biswas biswas PROPN cana-1326 251 17 : : PUNCT cana-1326 251 18 common common ADJ cana-1326 251 19 fixed fix VERB cana-1326 251 20 point point NOUN cana-1326 251 21 theorems theorem NOUN cana-1326 251 22 under under ADP cana-1326 251 23 rational rational ADJ cana-1326 251 24 contractions contraction NOUN cana-1326 251 25 for for ADP cana-1326 251 26 pair pair NOUN cana-1326 251 27 of of ADP cana-1326 251 28 mappings mapping NOUN cana-1326 251 29 in in ADP cana-1326 251 30 bicomplex bicomplex NOUN cana-1326 251 31 valued value VERB cana-1326 251 32 metric metric ADJ cana-1326 251 33 spaces space NOUN cana-1326 251 34 , , PUNCT cana-1326 251 35 journal journal NOUN cana-1326 251 36 of of ADP cana-1326 251 37 interdisciplinary interdisciplinary ADJ cana-1326 251 38 mathematics mathematic NOUN cana-1326 251 39 , , PUNCT cana-1326 251 40 vol.22(2019),no.7 vol.22(2019),no.7 PROPN cana-1326 251 41 , , PUNCT cana-1326 251 42 1071 1071 NUM cana-1326 251 43 - - SYM cana-1326 251 44 1082 1082 NUM cana-1326 251 45 . . PUNCT cana-1326 252 1 [ [ X cana-1326 252 2 9 9 NUM cana-1326 252 3 ] ] SYM cana-1326 252 4 a. a. NOUN cana-1326 252 5 azam azam PROPN cana-1326 252 6 , , PUNCT cana-1326 252 7 j. j. PROPN cana-1326 252 8 amad amad PROPN cana-1326 252 9 and and CCONJ cana-1326 252 10 p.kumam p.kumam NOUN cana-1326 252 11 : : PUNCT cana-1326 252 12 common common ADJ cana-1326 252 13 fixed fix VERB cana-1326 252 14 point point NOUN cana-1326 252 15 theorems theorem NOUN cana-1326 252 16 for for ADP cana-1326 252 17 multi multi ADJ cana-1326 252 18 - - ADJ cana-1326 252 19 valued value VERB cana-1326 252 20 mappings mapping NOUN cana-1326 252 21 in in ADP cana-1326 252 22 complex complex ADJ cana-1326 252 23 valued value VERB cana-1326 252 24 metric metric ADJ cana-1326 252 25 spaces space NOUN cana-1326 252 26 , , PUNCT cana-1326 252 27 j. j. PROPN cana-1326 252 28 inequal.appl inequal.appl PROPN cana-1326 252 29 . . PROPN cana-1326 252 30 ,2013(578 ,2013(578 PROPN cana-1326 252 31 ) ) PUNCT cana-1326 252 32 ( ( PUNCT cana-1326 252 33 2013 2013 NUM cana-1326 252 34 ) ) PUNCT cana-1326 252 35 . . PUNCT cana-1326 253 1 [ [ X cana-1326 253 2 10 10 NUM cana-1326 253 3 ] ] X cana-1326 253 4 i.a i.a PROPN cana-1326 253 5 . . PROPN cana-1326 253 6 bhahtim bhahtim PROPN cana-1326 253 7 : : PUNCT cana-1326 253 8 the the DET cana-1326 253 9 contraction contraction NOUN cana-1326 253 10 principle principle NOUN cana-1326 253 11 in in ADP cana-1326 253 12 quasi quasi ADJ cana-1326 253 13 metric metric ADJ cana-1326 253 14 spaces space NOUN cana-1326 253 15 , , PUNCT cana-1326 253 16 fuct.anal fuct.anal PROPN cana-1326 253 17 . . PUNCT cana-1326 253 18 ,30(1989),26 ,30(1989),26 PROPN cana-1326 253 19 - - PUNCT cana-1326 253 20 37 37 NUM cana-1326 253 21 . . PUNCT cana-1326 254 1 [ [ X cana-1326 254 2 11 11 NUM cana-1326 254 3 ] ] PUNCT cana-1326 254 4 c.klin c.klin NOUN cana-1326 254 5 - - PUNCT cana-1326 254 6 eam eam PROPN cana-1326 254 7 and and CCONJ cana-1326 254 8 c. c. PROPN cana-1326 254 9 suanoom suanoom NOUN cana-1326 254 10 : : PUNCT cana-1326 254 11 some some DET cana-1326 254 12 common common ADJ cana-1326 254 13 foxed fox VERB cana-1326 254 14 point point NOUN cana-1326 254 15 theorems theorem NOUN cana-1326 254 16 for for ADP cana-1326 254 17 generalized generalized ADJ cana-1326 254 18 contractive contractive ADJ cana-1326 254 19 type type NOUN cana-1326 254 20 mappings mapping NOUN cana-1326 254 21 on on ADP cana-1326 254 22 complex complex ADJ cana-1326 254 23 valued value VERB cana-1326 254 24 metric metric ADJ cana-1326 254 25 spaces space NOUN cana-1326 254 26 , , PUNCT cana-1326 254 27 abstr.appl.anal.2013 abstr.appl.anal.2013 ADJ cana-1326 254 28 ( ( PUNCT cana-1326 254 29 2013 2013 NUM cana-1326 254 30 ) ) PUNCT cana-1326 254 31 , , PUNCT cana-1326 254 32 article article NOUN cana-1326 254 33 i i PROPN cana-1326 254 34 d d PROPN cana-1326 254 35 604215 604215 NUM cana-1326 254 36 . . PUNCT cana-1326 255 1 [ [ X cana-1326 255 2 12 12 NUM cana-1326 255 3 ] ] X cana-1326 255 4 md.a.hoque md.a.hoque X cana-1326 255 5 : : PUNCT cana-1326 255 6 some some DET cana-1326 255 7 common common ADJ cana-1326 255 8 fixed fix VERB cana-1326 255 9 point point NOUN cana-1326 255 10 theorem theorem VERB cana-1326 255 11 in in ADP cana-1326 255 12 bicomplex bicomplex NOUN cana-1326 255 13 valued value VERB cana-1326 255 14 metric metric ADJ cana-1326 255 15 spaces space NOUN cana-1326 255 16 , , PUNCT cana-1326 255 17 submitted submit VERB cana-1326 255 18 for for ADP cana-1326 255 19 publication publication NOUN cana-1326 255 20 ( ( PUNCT cana-1326 255 21 2023 2023 NUM cana-1326 255 22 ) ) PUNCT cana-1326 255 23 . . PUNCT cana-1326 256 1 [ [ X cana-1326 256 2 13]j.ahmad 13]j.ahmad NUM cana-1326 256 3 , , PUNCT cana-1326 256 4 a.azam a.azam ADJ cana-1326 256 5 and and CCONJ cana-1326 256 6 s s PROPN cana-1326 256 7 saejung saejung PROPN cana-1326 256 8 : : PUNCT cana-1326 256 9 common common ADJ cana-1326 256 10 fixed fix VERB cana-1326 256 11 point point NOUN cana-1326 256 12 results result NOUN cana-1326 256 13 for for ADP cana-1326 256 14 contractive contractive ADJ cana-1326 256 15 mapping mapping NOUN cana-1326 256 16 in in ADP cana-1326 256 17 complex complex ADJ cana-1326 256 18 valued value VERB cana-1326 256 19 metric metric ADJ cana-1326 256 20 spaces space NOUN cana-1326 256 21 , , PUNCT cana-1326 256 22 fixed fix VERB cana-1326 256 23 point point NOUN cana-1326 256 24 theory theory NOUN cana-1326 256 25 and and CCONJ cana-1326 256 26 appl appl NOUN cana-1326 256 27 . . PROPN cana-1326 256 28 ,2014(67 ,2014(67 PROPN cana-1326 256 29 ) ) PUNCT cana-1326 256 30 ( ( PUNCT cana-1326 256 31 2014 2014 NUM cana-1326 256 32 ) ) PUNCT cana-1326 256 33 . . PUNCT cana-1326 257 1 [ [ X cana-1326 257 2 14 14 NUM cana-1326 257 3 ] ] X cana-1326 257 4 i. i. PROPN cana-1326 257 5 a. a. PROPN cana-1326 257 6 bakhtin bakhtin PROPN cana-1326 257 7 , , PUNCT cana-1326 257 8 β€œ " PUNCT cana-1326 257 9 the the DET cana-1326 257 10 contraction contraction NOUN cana-1326 257 11 principle principle VERB cana-1326 257 12 in in ADP cana-1326 257 13 quasi quasi ADJ cana-1326 257 14 metric metric ADJ cana-1326 257 15 spaces space NOUN cana-1326 257 16 , , PUNCT cana-1326 257 17 ” " PUNCT cana-1326 257 18 journal journal NOUN cana-1326 257 19 of of ADP cana-1326 257 20 functional functional ADJ cana-1326 257 21 analysis analysis NOUN cana-1326 257 22 , , PUNCT cana-1326 257 23 vol vol NOUN cana-1326 257 24 . . PROPN cana-1326 257 25 30 30 NUM cana-1326 257 26 , , PUNCT cana-1326 257 27 pp pp ADJ cana-1326 257 28 . . PUNCT cana-1326 258 1 26–37 26–37 NUM cana-1326 258 2 , , PUNCT cana-1326 258 3 1989 1989 NUM cana-1326 258 4 . . PUNCT cana-1326 259 1 [ [ X cana-1326 259 2 15 15 NUM cana-1326 259 3 ] ] X cana-1326 259 4 s. s. PROPN cana-1326 259 5 banach banach PROPN cana-1326 259 6 , , PUNCT cana-1326 259 7 β€œ " PUNCT cana-1326 259 8 sur sur PROPN cana-1326 259 9 les les X cana-1326 259 10 operations operation NOUN cana-1326 259 11 dans dan NOUN cana-1326 259 12 les le NOUN cana-1326 259 13 ensembles ensemble NOUN cana-1326 259 14 abstraits abstrait NOUN cana-1326 259 15 et et PROPN cana-1326 259 16 leur leur X cana-1326 259 17 application application PROPN cana-1326 259 18 aux aux PROPN cana-1326 259 19 equations equation NOUN cana-1326 259 20 integrals integral NOUN cana-1326 259 21 , , PUNCT cana-1326 259 22 ” " PUNCT cana-1326 259 23 fundamenta fundamenta PROPN cana-1326 259 24 mathematicae mathematicae PROPN cana-1326 259 25 , , PUNCT cana-1326 259 26 vol vol NOUN cana-1326 259 27 . . PROPN cana-1326 260 1 3 3 NUM cana-1326 260 2 , , PUNCT cana-1326 260 3 pp pp ADJ cana-1326 260 4 . . PUNCT cana-1326 261 1 133 133 NUM cana-1326 262 1 [ [ X cana-1326 262 2 16 16 NUM cana-1326 262 3 ] ] PUNCT cana-1326 262 4 a. a. NOUN cana-1326 262 5 a. a. NOUN cana-1326 262 6 mukheimer mukheimer PROPN cana-1326 262 7 , , PUNCT cana-1326 262 8 β€œ " PUNCT cana-1326 262 9 some some DET cana-1326 262 10 common common ADJ cana-1326 262 11 fixed fix VERB cana-1326 262 12 point point NOUN cana-1326 262 13 theorems theorem NOUN cana-1326 262 14 in in ADP cana-1326 262 15 complex complex ADJ cana-1326 262 16 valued value VERB cana-1326 262 17 b b X cana-1326 262 18 - - PUNCT cana-1326 262 19 metric metric ADJ cana-1326 262 20 spaces space NOUN cana-1326 262 21 , , PUNCT cana-1326 262 22 ” " PUNCT cana-1326 262 23 the the DET cana-1326 262 24 scientific scientific ADJ cana-1326 262 25 world world NOUN cana-1326 262 26 journal journal NOUN cana-1326 262 27 , , PUNCT cana-1326 262 28 vol vol NOUN cana-1326 262 29 . . PROPN cana-1326 262 30 2014 2014 NUM cana-1326 262 31 , , PUNCT cana-1326 262 32 article article NOUN cana-1326 262 33 i i PROPN cana-1326 262 34 d d PROPN cana-1326 262 35 587825 587825 NUM cana-1326 262 36 , , PUNCT cana-1326 262 37 6 6 NUM cana-1326 262 38 pages page NOUN cana-1326 262 39 , , PUNCT cana-1326 262 40 2014 2014 NUM cana-1326 262 41 . . PUNCT cana-1326 263 1 [ [ X cana-1326 263 2 17 17 NUM cana-1326 263 3 ] ] PUNCT cana-1326 263 4 a. a. PROPN cana-1326 263 5 k. k. PROPN cana-1326 263 6 dubey dubey PROPN cana-1326 263 7 , , PUNCT cana-1326 263 8 r. r. PROPN cana-1326 263 9 shukla shukla PROPN cana-1326 263 10 , , PUNCT cana-1326 263 11 and and CCONJ cana-1326 263 12 r. r. PROPN cana-1326 263 13 p. p. PROPN cana-1326 263 14 dubey dubey PROPN cana-1326 263 15 , , PUNCT cana-1326 263 16 β€œ " PUNCT cana-1326 263 17 some some DET cana-1326 263 18 fixed fix VERB cana-1326 263 19 point point NOUN cana-1326 263 20 theorems theorem NOUN cana-1326 263 21 in in ADP cana-1326 263 22 complex complex ADJ cana-1326 263 23 valued value VERB cana-1326 263 24 b b X cana-1326 263 25 - - PUNCT cana-1326 263 26 metric metric ADJ cana-1326 263 27 spaces space NOUN cana-1326 263 28 , , PUNCT cana-1326 263 29 ” " PUNCT cana-1326 263 30 journal journal NOUN cana-1326 263 31 of of ADP cana-1326 263 32 complex complex ADJ cana-1326 263 33 systems system NOUN cana-1326 263 34 , , PUNCT cana-1326 263 35 vol vol NOUN cana-1326 263 36 . . NOUN cana-1326 263 37 2015 2015 NUM cana-1326 263 38 , , PUNCT cana-1326 263 39 article article NOUN cana-1326 263 40 i i PROPN cana-1326 263 41 d d PROPN cana-1326 263 42 832467 832467 NUM cana-1326 263 43 , , PUNCT cana-1326 263 44 7 7 NUM cana-1326 263 45 pages page NOUN cana-1326 263 46 , , PUNCT cana-1326 263 47 2015 2015 NUM cana-1326 263 48 . . PUNCT cana-1326 264 1 [ [ X cana-1326 264 2 18 18 NUM cana-1326 264 3 ] ] PUNCT cana-1326 264 4 k. k. PROPN cana-1326 265 1 p. p. PROPN cana-1326 265 2 r. r. PROPN cana-1326 265 3 rao rao PROPN cana-1326 265 4 , , PUNCT cana-1326 265 5 p. p. PROPN cana-1326 265 6 r. r. PROPN cana-1326 265 7 swamy swamy PROPN cana-1326 265 8 , , PUNCT cana-1326 265 9 and and CCONJ cana-1326 265 10 j. j. PROPN cana-1326 265 11 r. r. PROPN cana-1326 265 12 prasad prasad PROPN cana-1326 265 13 , , PUNCT cana-1326 265 14 β€œ " PUNCT cana-1326 265 15 a a DET cana-1326 265 16 common common ADJ cana-1326 265 17 fixed fix VERB cana-1326 265 18 point point NOUN cana-1326 265 19 theorem theorem VERB cana-1326 265 20 in in ADP cana-1326 265 21 complex complex ADJ cana-1326 265 22 valued value VERB cana-1326 265 23 b b X cana-1326 265 24 - - PUNCT cana-1326 265 25 metric metric ADJ cana-1326 265 26 spaces space NOUN cana-1326 265 27 , , PUNCT cana-1326 265 28 ” " PUNCT cana-1326 265 29 bulletin bulletin NOUN cana-1326 265 30 of of ADP cana-1326 265 31 mathematics mathematic NOUN cana-1326 265 32 and and CCONJ cana-1326 265 33 statistics statistic NOUN cana-1326 265 34 research research NOUN cana-1326 265 35 , , PUNCT cana-1326 265 36 vol vol NOUN cana-1326 265 37 . . PROPN cana-1326 266 1 1 1 NUM cana-1326 266 2 , , PUNCT cana-1326 266 3 no no INTJ cana-1326 266 4 . . NOUN cana-1326 266 5 1 1 NUM cana-1326 266 6 , , PUNCT cana-1326 266 7 pp pp ADJ cana-1326 266 8 . . PUNCT cana-1326 267 1 1–8 1–8 NUM cana-1326 267 2 , , PUNCT cana-1326 267 3 2013 2013 NUM cana-1326 267 4 . . PUNCT