real analysis - Prove that if $lim limits_{n to infty}$ $x_n$ = $L$, then $lim limits_{n to infty}$ $|x_n|$ = $|L|$.

This question has two other parts.

b) Give an example of a sequence $(x_n)$n∈$\mathbb{N}$ such that (|$x_n$
|)n∈$\mathbb{N}$ converges, but ($x_n$)n∈$\mathbb{N}$ diverges.

c) Prove that if $\lim \limits_{n \to \infty}$ $|x_n|$ = 0, then $\lim \limits_{n \to \infty}$ $x_n$ = 0.

I'm pretty sure we're supposed to use our definition for convergences, which says that Given a real number $L$, we say that $(X_n)$ converges to L if for every $\epsilon$>0, there exists N∈$\Bbb{N}$ such that for all n∈N satisfying n>N, we have |$X_n-L$|<$\epsilon$.

Answer

For the first part, just use the "left-sided" triangle inequality in the $\delta-\epsilon$ definition.

For part b), the classical example is the sequence $a_n=(-1)^n$.

Part c) is trivial given $||x_n||=|x_n|<\epsilon$ implies convergence.