Proof that $\lim_{n\to\infty} a_n = L$ then $\lim_{n\to\infty} |a_n| = |L|$

Question

In order to prove that

$$\lim_{n\to\infty} a_n = L \implies \lim_{n\to\infty} |a_n| = |L|$$

I started with my assumption:

$\lim_{n\to\infty} a_n = L$ we have:

$$n>n_0\implies |a_n - L| < \epsilon $$

but I can say that $$||a_n|-|L||<|a_n -L| < \epsilon$$ rigth?

Then my proof is done?

Answer 1

Just a small technicality: you should write $||a_n| - |L|| \color{red}{\le} |a_n - L| < \epsilon$, for it may be the case that $a_n = L$ for some $n$.