Middle European Mathematical Olympiad 2025 Problem I-1

Let $\mathbb{R}^+$ be the set of positive real numbers. Let $f\colon \mathbb{R}^{+}\to \mathbb{R}^{+}$ be a function such that for all $x,y\in \mathbb{R}^{+}$ it holds that

$$y f^{2025}(x) \geq x f(y).$$

Show that there exists a positive integer $n_0$ such that for all positive integers $n \geq n_0$ and for all $x \in \mathbb{R}^+$ it holds that

$$f^n(x) \geq x.$$

Remark. Here $f^n$ denotes the function $f$ applied $n$ times, this means $f^n(x) = \underbrace{f(f(\ldots f(x)\ldots))}_{n \text{ times}}$.