Prove that if $a_n \geq 0, b_n > 0$ for $n \in \mathbb{N} $ , $\sum_{n=0}^\infty b_n$ is convergent and $\lim_{n\rightarrow\infty} \frac{a_n}{b_n} = 0$ then $\sum_{n=0}^\infty a_n$ is also convergent.
What I did was let $\sum_{n=0}^\infty a_n$ be divergent then $\lim_{n\rightarrow\infty} \frac{a_n}{b_n} = 0$ is true if $\lim_{n\rightarrow\infty}b_n = \infty$ which is false, because $\sum_{n=0}^\infty b_n$ is convergent, SO $\lim_{n\rightarrow\infty} a_n$ = 0 which is a contradiction to $\sum_{n=0}^\infty a_n$ being divergent.
My question: Is this correct and if not how can I fix it or solve the problem some other way?