Ok so the question is:

Prove carefully that a convergent sequence is bounded.

Here is my proof thus far, but i'm not really sure if it is correct\finished:

$\displaystyle |a_n| = |a_n - L + L| \leq |a_n - L| + |L|$

$\displaystyle a_n$ is convergent, say to a limit L so:

Let $\displaystyle \epsilon > 0$

$\displaystyle \exists N \in \mathbb{N}$ such that $\displaystyle \forall n \geq N |a_n - L| < \epsilon$.

$\displaystyle |a_n| \leq |a_n - L| + |L| < \epsilon + |L|$ $\displaystyle \forall n \geq N$.

Is this enough to show that it is bounded?