Prove, by definition, that if $\displaystyle \lim_{n\to \infty} a_n = L$ then $\displaystyle \lim_{n\to \infty} a_n^2 = L^2$.

It should be a basic proof but I'm somehow stuck. I was obviously thinking:

$\displaystyle \mid a_n^2 - L^2 \mid = \mid a_n - L \mid \cdot \mid a_n + L\mid$

but I'm not sure how to proceed.

Any help would be appreciated.