Limit comparison test proof

We had the following theorem in class:

Let $(a_n)$ and $(b_n)$ be sequences and $b_n>0$ and $\lim_{n\rightarrow\infty}\frac{a_n}{b_n}=L$ with $L\in\mathbb R\backslash\{0\}$. Then $\sum a_n$ converges if and only if $\sum b_n$ converges.

So by recapitulating the lecture I’ve tried to prove it but I didn’t get it. It’s obvious that you have to use the comparison test, but how? Can anybody help? Thanks a lot!

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