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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 $a=0$.2012-03-23

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Hint: Look at the sequence $1,-1,1,-1,\ldots$.

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    Note that you have to impose that $lim a_n$ doesn't exist!2012-03-23
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    do you mean choose (a_n)= ((-1)^n) ?2012-03-23
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    @user27217 Yes, exactly.2012-03-23
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    thanks, it suits now. simple things sometimes being hard to see2012-03-23
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    @chessmath: Actually, the converse is false even if $\lim a_n$ does exist: consider $a_n=1$ for all $n$, but $a=-1$.2012-03-23
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    But u have to keep in mind that it holds when a=0, and when a is not zero the sequence will have a signal for n large enough!2012-03-23