If $a_n,b_n\geq 0,$ then does $\lim_{n\to\infty} \frac{a_n}{b_n} = \gamma \implies \lim_{n\to\infty} \left(\frac{\sum a_n}{\sum b_n}\right)=\gamma\ ?$

Question

More precisely, if $\ (a_n)\ $ and $\ (b_n)\ $ are non-negative real sequences with $\ a_n\not\to 0,\ b_n\not\to 0\ $ and $\ \displaystyle\lim_{n\to\infty} \left( \frac{a_n}{b_n}\right) = \gamma,$ then is it true that $\ \displaystyle\lim_{n\to\infty} \left(\frac{\displaystyle\sum_{i=1}^{n} a_i}{\displaystyle\sum_{i=1}^{n} b_i}\right)=\gamma\quad ?$

This feels like it should be true, but not sure if there is a straightforward way to prove it.