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?
Answer
Hint
A convergent sequence is bounded. So you can also bound $\vert a+a_n\vert$.
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