Problem

Source: China TST 1996, problem 2

Tags: function, limit, induction, strong induction, algebra unsolved, algebra



$S$ is the set of functions $f:\mathbb{N} \to \mathbb{R}$ that satisfy the following conditions: I. $f(1) = 2$ II. $f(n+1) \geq f(n) \geq \frac{n}{n + 1} f(2n)$ for $n = 1, 2, \ldots$ Find the smallest $M \in \mathbb{N}$ such that for any $f \in S$ and any $n \in \mathbb{N}, f(n) < M$.