Simple Question About Bounded And Convergent Sequence

Given $\{a_n\}$ sequence which is bounded by: $m\le a_n \le M$ and converges to $L\in\mathbb{R}$. How do I prove that $m\le L \le M$?

Thank you very much.

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HINT: Suppose that $L>M$. Let $\epsilon=L-M$. Then there is an $n_0$ such that $L-\epsilon<a_n<L+\epsilon$ whenever $n\ge n_0$. But then ... what?
I get that $M<a_n$ which is a contradiction; thank you very much . :) – Anonymous Apr 30 '12 at 17:35
You could also notice that $[m,M]$ is a closed set, and $a_n \in [m,M]$, for all $n$. Hence the limit must also be in $[m,M]$.