# Using the Sandwich Theorem

• Feb 5th 2013, 02:55 PM
sakuraxkisu
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!
• Feb 5th 2013, 03:08 PM
Plato
Re: Using the Sandwich Theorem
Quote:

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.
• Feb 5th 2013, 11:49 PM
sakuraxkisu
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 :)
• Feb 6th 2013, 03:46 AM
Plato
Re: Using the Sandwich Theorem
Quote:

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.
• Feb 6th 2013, 05:26 AM
emakarov
Re: Using the Sandwich Theorem
Quote:

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 $\displaystyle c_n\le |a_nb_n|\le d_n$.