Question 25
Let $f, g$ be two functions from the set $\mathbf{R}$ of real numbers to itself, such that $f(x)\lt g(x)$ for all $x \in \mathbf{R}$. Prove that there exists an infinite subset $S \subseteq \mathbf{R}$ such that $f(x)\lt g(y)$ for all $x, y \in S$.