[Math] Prove if $a_n$ converges to $0$ and $b_n$ is bounded, then $a_n b_n$ converges to $0$

Question

We have these two hypothesis:

$$\forall\epsilon_1>0, \exists n_0 | n>n_o \implies |a_n|<\epsilon_1$$
$$|b_n|<M$$
where $M$ is the sequence bound.

Therefore, I've used hypothesis 2 to multiply both sides in the hypothesis 1 so we have:

$$\forall\epsilon_1>0, \exists n_0 | n>n_o \implies |a_n b_n|<\epsilon_1M$$

then if we choose $\epsilon = \epsilon_1 M$ we have:

$$\forall\epsilon>0, \exists n_0 | n>n_o \implies |a_n b_n|<\epsilon$$

Am I right?

Answer 1

Hint: Take any $\epsilon >0$ then $|a_n| < \frac{\epsilon}{M}$, whenever $n> n_0$, for some $n_0 \in \mathbb N$, where $M$ is such that $|b_n|\leq M$ then

$$|a_n b_n|= |a_n||b_n|<\frac{\epsilon}{M} M = \epsilon $$