- #1

sitia

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## Homework Statement

Consider sequences {an} and {bn}, where sequence {bn} converges to 0.

Is it true that sequence {anbn} converges to 0?

## 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!