Prove that if the absolute value of a sequence converges to $0$, the sequence converges to $0$ as well. Let ($a_n)_{n \in \mathbb{N}}$ be a sequence, prove that if $|a_n|$ converges to $0$ then ($a_n)_{n \in \mathbb{N}}$ converges to 0 as well.
Now let $|a_n|$ converge to $0$ and let $\epsilon > 0$ that means that  $||a_n|-0| < \epsilon$ for an $N \in \mathbb{N}$ such that $n > N$.
But how do I go from here? I should conclude from  $||a_n|-0| < \epsilon$ that  $|a_n-0| < \epsilon$.

You're almost there:
$||a_n|-0| = ||a_n|| = |a_n| = |a_n-0|$
