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.
How to prove $\lim|a_n|=|a|$ does not mean $\lim a_n =a$ exist
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real-analysis
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0Note that the converse is always true iff $a=0$. – 2012-03-23
1 Answers
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Hint: Look at the sequence $1,-1,1,-1,\ldots$.
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0But 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