What is needed for the inequality $c_1\cdot r^n \le f(r,n) \le c_2\cdot r^n$ to imply that $f(r,n)$ is of the form $c\cdot r^n$?

Question

My conjuncture is that the key is that the inequality holds for abitrary small/big $r$, and therefore the difference between $$\frac{f(r,n)}{r^n}$$ and $c_1$ or $c_2$ is bounded.

I don't quite see though how to proceed from here.