analysis - If $frac{a_{n+1}}{a_n} leq frac{b_{n+1}}{b_n}$ for all $n$, prove series properties

$a_n$ and $b_n$ are sequences of non-negative terms

$\dfrac{a_{n+1}}{a_n} \leq \dfrac{b_{n+1}}{b_n}$ for all $n$

If $\sum\limits_{n=1}^\infty b_n$ converges, prove that $\sum\limits_{n=1}^\infty a_n$ converges

If $\sum\limits_{n=1}^\infty a_n$ diverges, prove that $\sum\limits_{n=1}^\infty b_n$ diverges

I was thinking of trying the ratio test. For the 1st part, $\sum b_n$ converges, which implies that $\lim\limits_{n\to\infty}\left(\dfrac{b_{n+1}}{b_n}\right)$ is less than or equal to $1$. If it is less than $1$, than the same limit for $\frac{a_{n+1}}{a_n}$ is less than one, which implies $\sum\limits_{n=1}^\infty a_n$ converges. But I am not sure what to do about if the limit equals $1$.

Thanks for the help