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December 6th, 2010, 11:26 AM   #1
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proof of convergence....please help!

Problem:
Prove if {bn} converges to B and B ? 0 and bn ? 0 for all n, then there is M>0 such that |bn|?M for for all n.

What I have so far:
I know that if {bn} converges to B and B ? 0 then their is a positive real number M and a positive integer N such that if n?N, then |bn|?M . (by lemma)
PROOF (of lemma)- since B ? 0, (|B|)/2=E>0. There is N such that if n?N, then
|bn-B|<E. Let M=[(|B|)/2]. thus for n?N,
|bn|=|bn-B+B|?|B|-|bn-B|?|B|-[(|B|)/2]=[(|B|)/2]=M

what variations do i need to make in the proof for the lemma?
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December 6th, 2010, 12:46 PM   #2
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Re: proof of convergence....please help!

That's pretty much the right idea.
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December 6th, 2010, 12:58 PM   #3
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Re: proof of convergence....please help!

i know that it is similar but there are some differences......im just not sure what they are
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December 6th, 2010, 01:35 PM   #4
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Re: proof of convergence....please help!

The only diffrence is a general quantifier placed in proposed problem. After proving this lemma it is almost done- consider such that , where are intiger and real from lemma. The remaining sequence elements modules are of course greater then [latex]M>M'[latex].
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