14. (a) Suppose (x) is a sequence of points in R latisfying a s x sb...

 

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14. (a) Suppose (x) is a sequence of points in R latisfying a s x sb for all k e N. Prove that (x) has a convergent subsequence terms in the sequence, this should be easy. If there are infinitely many distinct terms in the sequence, then there must be infinitely many either in the left half-interval [a, t) or in the right half-interval [, b). Let (a, bi] be such a half-interval. Continue the process, and apply Exercise 10.) (b) Use the results of Exercises 12 and 13 to prove that any Cauchy sequence in R is convergent. (e) Now prove that any Cauchy sequence in R" is convergent. (Hint: If there are only finitely many distinct 14. (a) Suppose (x) is a sequence of points in R latisfying a s x sb for all k e N. Prove that (x) has a convergent subsequence terms in the sequence, this should be easy. If there are infinitely many distinct terms in the sequence, then there must be infinitely many either in the left half-interval [a, t) or in the right half-interval [, b). Let (a, bi] be such a half-interval. Continue the process, and apply Exercise 10.) (b) Use the results of Exercises 12 and 13 to prove that any Cauchy sequence in R is convergent. (e) Now prove that any Cauchy sequence in R" is convergent. (Hint: If there are only finitely many distinct

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LA a Sup pose fR is a Segnence q pomts in R b Prore that any cauchy stguenc... View the full answer