# Thread: Using the Sandwich Theorem

1. ## Using the Sandwich Theorem

Suppose that the sequence (a_n) converges to 0 and (b_n) is bounded. Use the Sandwich Theorem to prove that (a_n)(b_n) converges to 0.

I know that, since (b_n) is bounded, it is convergent to a limit L. But I'm not sure how to use the Sandwich Theorem to prove the product converges to 0 - any help would be greatly appreciated!

2. ## Re: Using the Sandwich Theorem

Originally Posted by sakuraxkisu
Suppose that the sequence (a_n) converges to 0 and (b_n) is bounded. Use the Sandwich Theorem to prove that (a_n)(b_n) converges to 0.
I know that, since (b_n) is bounded, it is convergent to a limit L. But I'm not sure how to use the Sandwich Theorem to prove the product converges to 0 - any help would be greatly appreciated!
That is a false claim.

It is true that $\displaystyle \exists B>0$ such that $\displaystyle (\forall n)[|b_n|\le B]~.$

If $\displaystyle \epsilon>0$ then $\displaystyle (\exists N>0)(\forall n\ge N)\left[|a_n|<\frac{\epsilon}{B}\right]$.

Now you finish.

3. ## Re: Using the Sandwich Theorem

Hi, thank you for your reply. Sorry about the mistake! I was wondering, would you perhaps be able to explain how the Sandwich Theorem can be used to prove this? Thanks

4. ## Re: Using the Sandwich Theorem

Originally Posted by sakuraxkisu
Hi, thank you for your reply. Sorry about the mistake! I was wondering, would you perhaps be able to explain how the Sandwich Theorem can be used to prove this?
There is no reason to use the Sandwich Theorem.
If you must, I don't see right now.

5. ## Re: Using the Sandwich Theorem

Originally Posted by sakuraxkisu
I was wondering, would you perhaps be able to explain how the Sandwich Theorem can be used to prove this?
Consider $\displaystyle c_n=0$ and $\displaystyle d_n=B|a_n|$ where $\displaystyle |b_n|\le B$ for all n. Show that $\displaystyle c_n\le |a_nb_n|\le d_n$.