# Homework Help: Convergence proof

1. Sep 20, 2011

### applied

Hi,

I'm doing some homework from my analysis class. I honestly have no idea where to start. Any help would be appreciated.

1. The problem statement, all variables and given/known data

Let ${a_n}$ be a sequence that converges to 0, and let ${b_n}$ be a sequence. Prove that the sequence $a_n b_n$ converges to 0.

2. Relevant equations

3. The attempt at a solution

2. Sep 20, 2011

### Robert1986

If b is the limit of $b_n$ then you can make the terms of $b_n$ as close to b as you want by making n big enough. Also you can make a_n as close to 0 as you like by making n big enough. Now, if $\epsilon > 0$ you want to make the terms of $a_nb_m$ less than $\epsilon$ for large enough n. Now, using what I mentioned above, how can you find an n big enough?

3. Sep 20, 2011

### estro

$a_n=\frac{1}{n}$, $b_n=n^2$
$a_n b_n=n$

$\lim_{n\rightarrow \infty} a_n=0$
$\lim_{n\rightarrow \infty} a_n b_n \not= 0$

Last edited: Sep 20, 2011
4. Sep 20, 2011

### applied

I need the proof using epsilon. The prof. wants description of every step. its a senior level class

5. Sep 20, 2011

### applied

no idea

6. Sep 20, 2011

### estro

Try looking more closely at what I wrote.

7. Sep 20, 2011

### applied

it shows, it does not converge

8. Sep 20, 2011

### Robert1986

Whoops! I didn't read the problem carefully. Estro is correct; it does not, in general, converge to anything, much less 0. My apologies!

9. Sep 20, 2011

### gb7nash

Did you copy the problem correctly? If you did, whoever made the problem up made an error.