Convergence Proof of Sequence a_n b_n to 0

[applied]

Hi,

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

Homework Statement

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.

Homework Equations

The Attempt at a Solution

 

[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?

 

[estro]

...
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.
...

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

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

 

[applied]

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

 

[applied]

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?

no idea

 

[estro]

Try looking more closely at what I wrote.

 

[applied]

it shows, it does not converge

 

[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!

 

[gb7nash]

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