(adsbygoogle = window.adsbygoogle || []).push({}); Analysis Proof (simple, but urgent!)

1. The problem statement, all variables and given/known data

This is an exercise from Rudin Chapter 3.

#8. If [tex]\sum a_{n}[/tex] converges, and if {bn} is monotonic and bounded, prove that [tex]\sum b_{n} a_{n}[/tex] converges.

2. Relevant equations

THEOREM: Suppose {sn} is monotonic. Then {sn} converges <==> {sn} is bounded.

3. The attempt at a solution

{bn} converges. bn --> b

So, for any [tex]\epsilon > 0 \exists N [/tex] such that n > N implies [tex]|b - b_{n}| < \epsilon[/tex]

Let M = b if b > bN

Let M = [tex]\epsilon + b[/tex] if b <= bN

Now, [tex]\sum M a_{n} = M \sum a_{n}[/tex] for M constant. And [tex]\sum M a_{n} [/tex] converges to Ma. So, [tex]\sum b_{n} a_{n}[/tex] converges.

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Am I on the right track with this? I need to turn this in soon and any help would be GREAT!

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# Homework Help: Analysis Proof (simple, but !)

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