Q:

Limit of absolute value of sequence

Prove that if $\lim \{a_n\}\to a$ then $\lim \{|a_n|\}\to|a|$. Is the converse true? 
I don't know where to start.  Should I break this into 2 cases? 

A:

By definition, if $$\lim_{n \rightarrow \infty} a_n=a$$ then for all $\varepsilon>0$ there exists a $N \in \mathbb{N}$ such that $n \geq N$ implies $|a_n-a| \leq \varepsilon$.
We may also verify case-by-case that $$\big||x|-|y|\big| \leq |x-y|$$ for all $x,y \in \mathbb{R}$.  In particular,$$\big||a_n|-|a|\big| \leq |a_n-a|.$$

Combining these yields the proof.