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.