(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If [tex]\Sigma[/tex]a_{n}is a convergent series and {b_{n}} is a monotonic and bounded sequence, then [tex]\Sigma[/tex] a_{n}b_{n}is a convergent series.

2. Relevant equations

3. The attempt at a solution

Since {b_{n}} is bounded, |b_{n}|<M for some M>0. Since [tex]\Sigma[/tex]a_{n}is a convergent series, we have that for every [tex]\epsilon[/tex]>0, there is some N>0 such that |A_{m}-A_{n}|<[tex]\epsilon[/tex]/M for all m>n>N. Thus, [tex]\sum_{k=n}^m[/tex] a_{k}b_{k}< M [tex]\sum^{m}_{k=n}[/tex] (A_{k}-A_{k-1}) < [tex]\epsilon[/tex].

And

[tex]

\Sigma [/tex] a_{k}b_{k}

is convergent.

Is this correct? If it is, then where did the monotonic behavior of {b_{n}} get put in the proof?

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# Homework Help: Rudin convergence proof

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