Prove that if the sequence $\,\{a_n\}\,$ converges to $A$, then $\,\{|a_n\}|\,$ converges to |A|. Also, is the converse true?
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1The second $,A,$ must be $,|A|,$ ... – DonAntonio Feb 13 '13 at 14:46
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Yes sorry, I changed it – Student Feb 13 '13 at 14:47
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1Hint. – David Mitra Feb 13 '13 at 14:47
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@DavidMitra didn't even think to use that. Thanks – Student Feb 13 '13 at 14:49
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1You're welcome. (For your other question, think about sequences with alternating signs.) – David Mitra Feb 13 '13 at 14:50
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1The converse is not true: take $a_n=(-1)^n$ as a conterexample – AndreasT Feb 13 '13 at 14:51
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@DavidMitra OK - so say I use an example of a sequence with alternating signs...do you think in an analysis class a proof would be necessary or simply justification that the converse is false? – Student Feb 13 '13 at 14:54
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1A counterexample is what is needed. Show that your sequence satisfies $|x_n|\rightarrow |A|$ but $(x_n)$ does not converge to $A$. This will show that the converse is false. – David Mitra Feb 13 '13 at 14:56
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@Student What is the difference between justification and proof? A justification that fails to prove, is not justifying ... – Hagen von Eitzen Feb 13 '13 at 16:00
