Proving Absolute Value Convergence of Sequence to A

[IntroAnalysis]

Homework Statement

If the absolute value of a sequence, a_n converges to absolute value of A, does sequence, a_n necessarily converge to A?

Homework Equations

convergence: a sequence { an}n=1-->infinity, converges to A є R (A is called the limit of the sequence) iff for all є > 0, there exists an N є Natural, for all n$\geq$ N (│an - A│< є ).

Also, know that │ │a│ - │b││ $\leq$ │a - b│

The Attempt at a Solution

I've been trying to find a counterexample, but so far I haven't been able to. Any suggestions on this proof?

 

[jbunniii]

Consider a sequence whose sign changes frequently.

 

[IntroAnalysis]

So if my sequence = {(-1)ⁿ($\frac{1}{n}$)= {-1, 1/2, -1/3, 1/4, -1/5, ...} This is converging to 0.

If you have the absolute value of a_n, it also converges to 0. Remove the absolute value and you get the same convergence point. Maybe I don't have the sequence you had in mind?

 

[LCKurtz]

Try making the absolute value sequence converge to something other than 0.

 

[Ray Vickson]

Try the sequence {1,-1,1,-1,1,-1,...}.

RGV

 

[IntroAnalysis]

Thanks, I finally get it. l (-1) l ^2 will converge to l 1l whereas without the absolute value this sequence never converges but bounces back and forth from -1 to 1.

Also, what is a signature file so that I can add some needed terms?

 

[LCKurtz]

Also, what is a signature file so that I can add some needed terms?

Click on the "My PF" tab at the top of the page and you will see where to edit your signature file.