Let $(a_n)$ be a sequence. If $(|a_n|)$ is convergent then $(a_n)$ has a convergent subsequence.

Question

How can I prove this proposition?

Let $(a_n)$ be a sequence. If $(|a_n|)$ is convergent then $(a_n)$ has a convergent subsequence.

If $(|a_n|)$ converges to $L$ then $\forall \epsilon>0, \exists n_0 \in \mathbb{N}$ where $n>n_0 \implies ||a_n|-L|<\epsilon$.

I got stuck here.

Answer 1

hint

There are either infinitely many non-negative elements of $a_n$ or infinitely many non-positive elements. Consider the subsequence consisting of these.

Answer 2

Use Bolzano Weierstrass theorem. Every bounded sequence have subsequence convergence.