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## Homework Statement

Let ##\{a_n\}## be a bounded sequence such that every convergent subsequence has limit ##L##. Prove that ##\lim_{n\to\infty}a_n = L##.

## Homework Equations

## The Attempt at a Solution

I'm not really understanding this problem. Isn't ##\{a_n\}## a subsequence of itself? So isn't it immediately the case that ##\lim_{n\to\infty}a_n = L## by the hypothesis?

EDIT: Nevermind. I notice now that it says every

*convergent*subsequence.