## anonymous one year ago (Introductory Real Analysis) I'm trying to prove that the product of a bounded sequence and a sequence that converges to zero, itself converges to zero. I have the basic idea to start but don't understand some of the work my Professor did. Could someone help?

1. anonymous

$\text{Given}\ x_n \rightarrow 0, \ |y_n| \leq M,$$\text{Prove that:} \ x_ny_n \longrightarrow 0$

2. anonymous

$\text{Given} \ x_n \rightarrow 0,$$(\forall \lambda > 0)(\exists \ J \in N)(\forall n>J), \ \ \ |x_n-c|<\lambda$$(\exists \ M > 0)(\forall n \in N), \ \ \ |y_n|\leq M$

3. anonymous

Applying the above to our problem: $|x_n - 0|<\lambda$$|x_n|<\lambda$

4. anonymous

My confusion is here: $(\exists \ J \ \epsilon \ N)(\forall n > J), \ \ \ |x_n|<\frac{\varepsilon}{M}$

5. anonymous

How does that last part come about? @Zarkon @phi

6. phi

you have $(\forall \lambda > 0)(\exists \ J \in N)(\forall n>J), \ \ \ |x_n-c|<\lambda$ and that is true for all lambda's it looks like they decided to let lambda be epsilon/M .

7. anonymous

I had to move on, but I'd love to come back to this question sometime for my own sake, I frankly still don't get it.