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Cauchy Sequences and Complete Metric Spaces Let’s rst consider two examples of convergent sequences in R Example 1 Let x n = 1 n p 2 for each n2N Note that each x. The Online Books Page Listing over 3 million free books on the Web Updated Thursday, May 6, 21. U u t i=1 jf i(x) f i(a)j2 v u u t i=1 2=n= p 2 = Hence his a continuous function from Rminto Rn 4010 Let fa ng2l1 Prove that fde ned by f(fb ng) = P 1 n=1 a nb n is a continuous realvalued function on l1 2 Solution Let fb ng2l1 and >0 Since fa ng2l1there exists some M such that ja nj.