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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 $a=0$. –  Dejan Govc Mar 23 '12 at 17:52

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up vote 7 down vote accepted

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! –  checkmath Mar 23 '12 at 17:04
    
do you mean choose (a_n)= ((-1)^n) ? –  Shamill Mar 23 '12 at 17:04
    
@user27217 Yes, exactly. –  David Mitra Mar 23 '12 at 17:06
    
thanks, it suits now. simple things sometimes being hard to see –  Shamill Mar 23 '12 at 17:08
    
@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

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