Q:

Does convergence of the absolute value of a sequence imply convergence of the sequence?

I should check whether {${a_n}$} converges to $a$ if $|a_n|$ converges to $|a|$.
I found this is false, but I'm trying to find out counterexample.
Does it make sense that if $|a_n|$ converges to $|a|$, $a_n$ converges to $a$ or $-a$?

A:

Take $a_n = (-1)^n$. Then $\lvert a_n \rvert \longrightarrow 1$, but $(a_n)$ diverges.

