Sequence proof on absolute value

Question

So have to prove if $| a_n|\to0$, then $a_k\to0$ given that $a_k$ is a sequence.

I was wodnering how I might prove this, I considered that whats inside the absolute value must be positive and expanding it out, but got nowhere.

Answer 1

Using the $\epsilon$-definition of the limit, you have that

$$\forall\epsilon >0:\exists n_0\in\mathbb{N}:||a_n|-0|<\epsilon\text{ for all }n\geq n_0$$

i.e., as $||a_n|-0|=||a_n||=|a_n|=|a_n-0|$, you have that

$$\forall\epsilon >0:\exists n_0\in\mathbb{N}:|a_n-0|<\epsilon\text{ for all }n\geq n_0$$

which is per definition $\lim_{n\to\infty}a_n=0$.