IMC 2000 · Problem 6

Let $f : \mathbb{R} \to (0,\infty)$ be an increasing differentiable function for which $\lim\limits_{x \to \infty} f(x) = \infty$ and $f'$ is bounded.

Let $F(x) = \int_0^x f$. Define the sequence $(a_n)$ inductively by

$$a_0 = 1, \quad a_{n+1} = a_n + \frac{1}{f(a_n)},$$

and the sequence $(b_n)$ simply by $b_n = F^{-1}(n)$. Prove that $\lim\limits_{n \to \infty} (a_n - b_n) = 0$.