# Real analysis: Limit of a product of sequences

1. Nov 15, 2011

### frenchkiki

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

Let (u$_{n}$)$_{n}$ be a real sequence such that lim u$_{n}$ = 0 as x→∞ and let (v$_{n}$)$_{n}$ be a bounded sequence. Show that lim (u$_{n}$)$_{n}$(v$_{n}$)$_{n}$ = 0 as x→∞

2. Relevant equations

3. The attempt at a solution

Since (v$_{n}$)$_{n}$ is bounded then it has a least upper bound and greatest lower bound. Then we have g.l.b< lim (v$_{n}$)$_{n}$ <l.u.b
I don't really know how to take it from here. Does the existence of the limit of (u$_{n}$)$_{n}$ mean it is bounded?

Last edited: Nov 15, 2011