Limits help

Hi everyone, I currently have this problem to prove.




I am aware of the product rule (that if there exist sequences An and Bn which both have limits, then the limit of AnBn is the product of the 2 limts).
Although, the question says that the sequence Bn is bounded. However, I know that a sequence which is bounded does not necessarily have a limit, e.g. (-1)^n.

Any help would be very appreciated!

You could just blindly start writing down epsilons. Have you tried writing down what it means for (a_n) to be a null sequence, and for (b_n) to be bounded?

For An to be a null sequence it means that there exists a NeN+ such that n> N => l an ( - 0 ) l < E

For Bn to be bounded it means there exists UeN and LeN such that for all values of n we have L < Bn < U

That is correct right?
That's all i've got wrote down, not sure how to advance with the problem from there

Yes. Can you state what you need to prove, in epsilon-terms?

We need to prove that lAnbn - 0 l < E?

For some E, yes. Let's fix an $\epsilon$. What you have so far is:

$\exists N: \forall \ n > N, \ |a_n| < \epsilon$
$\forall \ n > N, L < |b_n| < U$.

We want to show something about $|a_n b_n| = |a_n| |b_n|$; in particular, we want to bound it above by something arbitrarily small. Can you give me an upper bound?

So we have l AnBn l = l An l l Bn l < UE (as An is less than E and Bn is < U)?

But can't Bn be <0? and so if Bn < U that does not imply that l Bn l < U?

That bound is correct if you work only with the modulus of $b_n$ and defined $U > \text{all} \ |b_n|$.
$b_n$ can be less than 0, but $|b_n|$ is always less than $\text{max}(|U|, |L|)$. A more useful formulation of "b_n is bounded" is usually "$|b_n| < B$ for some $B$".

A more useful formulation of "b_n is bounded" is usually "$|b_n| < B$ for some $B$".

Ah yes now I understand that!

So what i'd do is state what I know (as we discussed at the start, what a null sequence is and what a bound is, and what we're trying to prove)

Then say as l An l < E and l Bn l < B, then we can say l AnBn - 0 l = l An l l Bn l < EB

which implies l AnBn l < EB and since B is a constant we can re-write this as l AnBn - 0 l < E and so we conclude AnBn is a null sequence?

Without wanting to be too pedantic, this should be really be something like:

A more useful formulation of "b_n is bounded" is usually "we can find B such that $|b_n| < B \, \forall n$.