Question

(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?

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

My confusion is here:

\[(\exists \ J \ \epsilon \ N)(\forall n > J), \ \ \ |x_n|<\frac{\varepsilon}{M}\]

Answer 5

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

Answer 6

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 .

Answer 7

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.