Given $\lim \limits_{n\to \infty}a_n=L$ and $\lim \limits_{n\to \infty}\frac{a_n}{b_n}=1$ then $\lim \limits_{n\to \infty}b_n=L$
I think that this claim is true:
$$\lim \limits_{n\to \infty}a_n=L$$
$$\lim \limits_{n\to \infty}\frac{a_n}{b_n}=1$$
using limits arithmetics I can say:
$$\lim \limits_{n\to \infty}\frac{a_n}{b_n}=\frac{\lim \limits_{n\to \infty}a_n}{\lim \limits_{n\to \infty}b_n}=1$$
$$\frac{\lim \limits_{n\to \infty}a_n}{\lim \limits_{n\to \infty}b_n}=1$$
$$\lim \limits_{n\to \infty}b_n\cdot 1=\lim \limits_{n\to \infty}a_n$$
$$\lim \limits_{n\to \infty}b_n=L.$$
is my proof correct?