How to prove $\lim|a_n|=|a|$ does not mean $\lim a_n =a$ exist

It is easy to prove that if $\lim a_n=a$ then $\lim|a_n|=|a|$ by using $||a_n |-|a||\le|a_n-a|$, but I can't show that the converse is false.

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Note that the converse is always true iff $a=0$. –  Dejan Govc Mar 23 '12 at 17:52

Hint: Look at the sequence $1,-1,1,-1,\ldots$.
Note that you have to impose that $lim a_n$ doesn't exist! –  checkmath Mar 23 '12 at 17:04
@chessmath: Actually, the converse is false even if $\lim a_n$ does exist: consider $a_n=1$ for all $n$, but $a=-1$. –  Jack Lee Mar 23 '12 at 18:50