Prove that $\limsup a_n b_n = a \limsup b_n$ given $a_n > 0$ and $\lim_{n \to \infty} a_n= a > 0$.

Let $a_n$ and $b_n$ be two sequences of real numbers. Assume that $a_n > 0$ and$\lim_{n \to \infty} a_n= a > 0$. Prove that $\limsup a_n b_n = a \limsup b_n$.

Actually, I know what to do if $(b_n)$ is bounded, but for the other case I have no idea.

Thank you for your time.