# Convergence of a series

1. Sep 11, 2012

### sitia

1. The problem statement, all variables and given/known data
Consider sequences {an} and {bn}, where sequence {bn} converges to 0.
Is it true that sequence {anbn} converges to 0?

3. The attempt at a solution

Proof.
First I assumed (an) is bounded, and so there exists M > 0 such that |an| < M for all n 2
{1, 2, 3, . . .}. Moreover, since lim bn = 0, there exists N such that, for all n≥ N,
|bn − 0| <ε /M
or, equivalently, |bn| < ε/M.
Therefore, whenever n ≥ N, we have that
|anbn − 0| = |anbn| = |an||bn| ≤M|bn| < M(ε/M)=ε

Since the choice of  ε> 0 was arbitrary, this implies that lim(anbn) = 0.

However, I know that {anbn} can also not converge if {an} is not bounded. Can someone help with how to go about that part of hte proof?
Thank you!

2. Sep 11, 2012

### susskind_leon

It obviously doesn't work when an is not bounded. I am understanding correctly that an does not converge to 0, right?
if so, just consider bn=1/n and an=n^2. an*bn=n does not convergen. Bam!

3. Sep 11, 2012

### sitia

There isn't given any information on if {an} converges to 0 or not. Thanks