Does $\lim_{n \to \infty} |a_n|=0$ and $\limsup_{n \to \infty} |b_n|<\infty$ imply $\lim_{n\to\infty} |a_nb_n|=0$?

Question

I would like to know if the following statement is true:

Let $(a_n)\subset \mathbb{C}$ and $(b_n)\subset \mathbb{C}.$ If $$\lim_{n \rightarrow \infty} |a_n|=0$$ and $$\limsup_{n \rightarrow \infty} |b_n|\leq \alpha<\infty,$$ then $$\lim_{n \rightarrow \infty}|a_n b_n|=0.$$

Thank you very much.

Masik

Answer 1

Yes, $\limsup |b_n|\le \alpha$ means that $|b_n|<\alpha+1$ eventually. This means that $0\le |a_nb_n|\le |a_n|(\alpha+1)$ eventually and the RHS $\to0$ as $n\to\infty$.

Answer 2

first of all we know that $lim_{n \to \infty} | b_n | <= M \ for M \in N$. there for if we assume that
so $ 0 <= lim_{n \to \infty} |a_n| \ lim_{n \to \infty}| b_n | <= lim_{n \to \infty} |a_n| \ M = 0 $.

Answer 3

First of all note that you can rewrite your problem assuming $(a_n)$ and $(b_n)$ are nonnegative real sequences, since you are only interested in the moduli of those sequences.

Hint: for a nonnegative sequence $(b_n)$, one has $$ \limsup b_n<\infty \qquad \text{if and only if } b_n \text{ is bounded}. $$ Can you proceed from here?