Prove: If $\lim_{n\rightarrow\infty}|a_n| = 0$, then $\lim_{n\rightarrow\infty}a_n = 0$

Question

Using the squeeze theorem, prove the following:

If $\lim_{n\rightarrow\infty}|a_n| = 0$, then $\lim_{n\rightarrow\infty}a_n = 0$.

Let $f(x) = |a_n|$. Let $g(x) = a_n$ such that $\forall x : g(x) \leq f(x)$. How do I continue this proof using the squeeze theorem? I want to construct a function $h(x)$ that lies in between those two functions.

Answer 1

Hint : $-|x|\le x\le |x|{}{}{}{}{}{}$

Answer 2

The accepted answer uses the squeeze Theorem.

Alternatively, note that $$ |a_n-0|=||a_n|-0|\tag{1} $$

Now if you unwrapping the definition of limits, the consequence is immediate by (1).