If $f$ is a strictly increasing function with $\lim_{x\rightarrow \infty}f(x)=\infty$ is it true that $\lim_{y\rightarrow \infty}f^{-1}(y)=\infty$?

Let $f:[0,\infty)\rightarrow \mathbb{R}_+$ be a strictly increasing continuous function with $f(0)=0$ and such that $\lim_{x\rightarrow \infty}f(x)=\infty$. I want to check if $\lim_{y\rightarrow \infty}f^{-1}(y)=\infty$, where $f^{-1}$ is the inverse of $f$.

My idea would be the following:

First let us remark that since $f$ is strictly increasing and continuous we indeed know that $f$ is invertible where $f^{-1}:[f(0),\infty)\rightarrow [0,\infty)$. From analysis we know that since $f$ is strictly increasing also $f^{-1}$ is strictly increasing.Since $\lim_{x\rightarrow \infty} f(x)=\infty$ and strictly increasing we know that for any $M>0$ there exists $x_M>M$ such that $f(x_M)>M$. Let $y=f(x_M)$ then $f^{-1}(y)=x_M>M$. Therefore we have shown that for any $M>0$ there exists $y>0$ such that $f^{-1}(y)>M$ which proves that $\lim_{y\rightarrow \infty} f^{-1}(y)=\infty$.

Does this work or if not could you show me how to do it?