Prove that $\lim _{n \to \infty}b_n=b$ and $\overline\lim_{n \to \infty}a_n=a$ then $\overline \lim a_nb_n=ab$

Question

Suppose $\{a_n\}, \{b_n\}$ are sequence of nonnegative real numbers with $\lim _{n \to \infty}b_n=b,b \neq 0$ and $\overline\lim_{n \to \infty}a_n=a$.

Then $\overline \lim a_nb_n=ab.$

let $\epsilon >0.$

Then $|a_nb_n-ab|=|a_nb_n-ab_n+ab_n-ab|=|b_n(a_n-a)+a(b_n-b)|\\ \le |b_n||a_n-a|+|a||b_n-b|.$

[Note: $\overline{\lim}$ is alternate notation for $\limsup.$]

How to processed from here?