Thread: Convergence with products of sequences

a) Let (an) be a bounded sequence (not necessarily convergent) and let (bn) be a sequence that converges to zero. Prove that the sequence of products (anbn) converges to zero.

b) Explain why you could not use the Algebraic Limit Theorem.

(My first thought was to use the ALT, so I'm confused. Not sure how to formally prove this, although it seems to make sense at the surface)

Originally Posted by Janu42

a) Let (an) be a bounded sequence (not necessarily convergent) and let (bn) be a sequence that converges to zero. Prove that the sequence of products (anbn) converges to zero.

b) Explain why you could not use the Algebraic Limit Theorem.

(My first thought was to use the ALT, so I'm confused. Not sure how to formally prove this, although it seems to make sense at the surface)

. Now take any ...

Tonio

Ok so that makes the product less than epsilon, showing it converges. How do I know that it converges to 0 though? And why is it that I can't use the ALT?

Originally Posted by Janu42

Ok so that makes the product less than epsilon, showing it converges. How do I know that it converges to 0 though? And why is it that I can't use the ALT?

.

I doubt that most of us even knows what ALT means.

Algebraic Limit Theorem, from part (B)

Originally Posted by Janu42

Algebraic Limit Theorem, from part (B)

That is not any help!
Why do you think that "Algebraic Limit Theorem" has any meaning to most people?

Why not state its statement?

Originally Posted by Janu42

Ok so that makes the product less than epsilon, showing it converges. How do I know that it converges to 0 though? And why is it that I can't use the ALT?

Uh? A sequence converges to zero iff for any we have for all the elements of the sequence but perhaps a finite number of them, and this is exactly what I showed above for ...

Tonio