1 i Let X = 1R and d be the usual metric on X that is, dix,y) = 1x-yl for each x y EX. Prove the followings monotone subsequence bounded sequence in t has a convergent i Every subsequence. X 1. Every sequence in t has Every Cauchy sequence in x is convergent.

Mathematics, Other Math

43 0

Get full Expert solution in seconds

$1.97 ONLY

1 i Let X = 1R and d be the usual metric on X that is, dix,y) = 1x-yl for each x y EX. Prove the followings monotone subseque

EXPERT ANSWER

x 82 casi continuing we get an increasing subsequence Every bounded sequence in x has a convergent Subsequence PH00f : if can

greatest en is test lowey bound of sequence xalth :? non sit lan-enke ln<l = an<lnte elte RN 2 9 2 Similauly · fom m)N have

L-€ con cun clt & if DN, en < xn < UN L- & can <lte أت 1 lan- LICE 3) lim an=1=U no