If a Cauchy sequence contains a convergent subsequence, must the Cauchy sequence have the same limit?

Asked Aug 25 '19 at 23:10

Active Aug 25 '19 at 23:12

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Let $X$ be a Banach space with norm $\|\cdot\|$. Let $(x_n)$ be a Cauchy sequence in $X$ containing a subsequence $(x_{n_k})$ which converges to some $x \in X$.

Is it true that $(x_n)$ must also converge to $x$?

I think the answer is yes, since $$\|x_n - x\| \leq \|x_n - x_{n_k}\| + \|x_{n_k} - x\| < \epsilon,$$ but I would appreciate any verification/correction.

edited Aug 25 '19 at 23:12

Xander Henderson

32,453

asked Aug 25 '19 at 23:10

kkc

1

For sake of contradiction, what if it didn't? – Pete Caradonna Aug 25 '19 at 23:15
One thing seems pretty clear: if it does converge, then it converges to $x$. – John Hughes Aug 25 '19 at 23:20
@JohnHughes well, $(x_n)$ is Cauchy in a complete space, so it must converge – kkc Aug 25 '19 at 23:21

If a Cauchy sequence contains a convergent subsequence, must the Cauchy sequence have the same limit?

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