TITLE: Prove/disprove: if $\lim\limits_{ n\to\infty} f(n)=\infty$ then $\lim\limits_{ n\to\infty}f(f(n))=\infty$
QUESTION [5 upvotes]: Let $f(x)$ a continuous function on $\Bbb{R}$.
Prove/disprove: If $\lim\limits_{n\to\infty} f(n)=\infty$, then $\lim\limits_{n\to\infty}f(f(n))=\infty,$ where the limits are taken over $n \in \mathbb N$. 

I have an hunch this statement isn't always true, but couldn't find a proper function to show this.

REPLY [2 votes]: When you say $f(n) \to \infty$ as $n \to \infty$, the usage of symbol $n$ usually means that $n$ is a positive integer (you have mentioned this explicitly also). Thus when the argument of $f$ tends to $\infty$ through positive integers then $f$ tends to $\infty$. In case of $f(f(n)) = f(t)$ the argument of $f$ is $t = f(n)$ and this itself may not be an integer. So even though the argument $t \to \infty$ it is not guaranteed to be an integer and hence $f(t)$ may not tend to $\infty$. If for all $n \in \mathbb{N}$ we ensure that $f(n) \in \mathbb{N}$ then $f(f(n)) \to \infty$ as $n \to \infty$.