Does $\lim_{n \to \infty} |a_n|=0$ and $\limsup_{n \to \infty} |b_n|<\infty$ imply $\lim_{n\to\infty} |a_nb_n|=0$? 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 $$\limsup_{n \rightarrow \infty} |b_n|\leq \alpha&lt;\infty,$$ then 
$$\lim_{n \rightarrow \infty}|a_n b_n|=0.$$

Thank you very much.

Masik
Yes, $\limsup |b_n|\le \alpha$ means that $|b_n|&lt;\alpha+1$ eventually. This means that $0\le |a_nb_n|\le |a_n|(\alpha+1)$ eventually and the RHS $\to0$ as $n\to\infty$.
first of all we know that 
$lim_{n \to \infty} | b_n | &lt;= M  \ for M \in N$.
there for if we assume that
so $ 0 &lt;= lim_{n \to \infty} |a_n| \ lim_{n \to \infty}| b_n | &lt;= lim_{n \to \infty} |a_n| \ M = 0 $.