Prove that $a_n \times b_n \to 0$ for $n \to \infty$

Question

I want to prove this example:

If $a_n \to 0$ for $n \to \infty$ and $(b_n)_n$ is bounded. Prove that $a_n \times b_n \to 0$ for $n \to \infty$.

My first guess is that I should use the definition of the boundedness and the convergence.

Therefore:

$|a_n| \leq M$ and $|a_n - a |< \epsilon$

My problem is, how to bring this two equations together to prove the theorem?

I appreciate your answer!!!

btw how to code in latex that the $n \to \infty$ is above the $\to$?

Answer 1

Hint: Let $M\in \mathbb{R}$ fulfil $|b_n|\leq M$ for all $n$. Then $$|a_n\cdot b_n|\leq |M|\cdot |a_n|$$