1. ## Sequence proof

Let $(a_{n})$ be a sequence which converges to 0, and let $(b_{n})$ be a sequence which is bounded but not necessarily convergent.

Prove that $lim (a_{n})(b_{n}) = 0$

No idea what to do here.

2. Well I am not so sure about this but cant you do something in this direction?

If $b_n$ is bounded then there exists M such that
$
|b_n|\leq M
$

For all $n\in \mathbb{N}$

Now then it is clear that:
$
\lim\limits_{n\to\infty}(a_n)(-M) \leq \lim\limits_{n\to\infty}(a_n)(b_n) \leq \lim\limits_{n\to\infty}(a_n)M
$

$
0\leq\lim\limits_{n\to\infty}(a_n)(b_n)\leq 0
$

Something like this?

3. I figured the squeeze theorem would show up in this one. I am horrible at generalizing like that. Thanks

4. Not a problem, I am not so good either.