(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Theorem: In a metric space X, if (x_{n}) is a Cauchy sequence with a subsequence (x_{n_k}) such that x_{n_k}-> a, then x_{n}->a.

2. Relevant equations

N/A

3. The attempt at a solution

1) According to this theorem, if we can show that ONE subsequence of x_{n}converges to a, is that enough? Or do we need to show that EVERY subseuqence of x_{n}converges to a in order to claim that x_{n}->a?

2) How can we prove the theorem?

By definition, x_{n}is Cauchy iff for all ε>0, there exists N s.t. if n,m≥N, then d(a_{n},a_{m})<ε.

By definition, x_{n}->a iff for all ε>0, there exists M s.t. if n≥M, then d(x_{n},a)<ε.

I know the definitions, but I don't see how to link these all together to prove the theorem...

Any help is greatly appreciated!

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# Homework Help: Cauchy sequence with a convergent subsequence

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