Real analysis: Limit of a product of sequences

[frenchkiki]

Homework Statement

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→∞

Homework Equations

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?

 

[mighty2000]

No, the existence of the limit of (u_{n})_{n} does not necessarily mean it is bounded. To show that lim (u_{n})_{n}(v_{n})_{n} = 0 as x→∞, we can use the definition of the limit. Let ε>0 be given. Since lim u_{n} = 0 as x→∞, there exists N such that |u_{n}|<ε for all n>N. Additionally, since (v_{n})_{n} is bounded, there exists M such that |v_{n}|<M for all n. Then, for n>N, we have |(u_{n})_{n}(v_{n})_{n}|<εM. Since εM is arbitrarily small, we can conclude that lim (u_{n})_{n}(v_{n})_{n} = 0 as x→∞.