Proving that if $\lim_{n \rightarrow \infty} a_n = a$ then $\lim_{n \rightarrow \infty} a_n^2 = a^2$ If we have a real sequence $\left|a_n\right|$ such that $\lim_{n \rightarrow \infty} a_n = a$, how do we prove (by an $\epsilon - N$ argument) that  $\left|a_n\right|$ such that $\lim_{n \rightarrow \infty} a_{n}^{2} = a^2$?
I know you can use algebra to do to the following:
$$\left|a_n^2 - a^2\right| =\left|(a_n - a)(a_n + a)\right|$$
Where I feel like you can use the implication that $\lim_{n \rightarrow \infty} a_n = a$ to show that $(a_n-a) < a$ or something.
What's the proper way to go about this? 

If for every $\varepsilon > 0$ there is some $N \geq 1$ such that $|a_{n}-a| < \varepsilon$, then, taking any $\varepsilon > 0$, there is some $N \geq 1$ such that
$$
|a_{n}^{2}-a^{2}| = |a_{n}-a||a_{n}+a| < 2|a|\varepsilon+\varepsilon^{2}
$$
for all $n \geq N$.
