MEMO 2024 I1

Determine all $k \in \mathbb{N}_0$ for which there exists a function $f: \mathbb{N}_0 \to \mathbb{N}_0$ such that $f(2024) = k$ and

$$f(f(n)) \leq f(n+1) - f(n)$$

for all $n \in \mathbb{N}_0$.

Remark. Here $\mathbb{N}_0$ denotes the set of nonnegative integers.