Let $\displaystyle (a_{n})$ be a sequence which converges to 0, and let $\displaystyle (b_{n})$ be a sequence which is bounded but not necessarily convergent.
Prove that $\displaystyle lim (a_{n})(b_{n}) = 0$
No idea what to do here.
Well I am not so sure about this but cant you do something in this direction?
If $\displaystyle b_n$ is bounded then there exists M such that
$\displaystyle
|b_n|\leq M
$
For all $\displaystyle n\in \mathbb{N}$
Now then it is clear that:
$\displaystyle
\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
$
$\displaystyle
0\leq\lim\limits_{n\to\infty}(a_n)(b_n)\leq 0
$
Something like this?