Školjka

%V0 Let $S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $(f, g)$ of functions from $S$ into $S$ is a Spanish Couple on $S$, if they satisfy the following conditions: (i) Both functions are strictly increasing, i.e. $f(x) < f(y)$ and $g(x) < g(y)$ for all $x$, $y\in S$ with $x < y$; (ii) The inequality $f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $x\in S$. Decide whether there exists a Spanish Couple on the set $S = \mathbb{N}$ of positive integers; on the set $S = \{a - \frac {1}{b}: a, b\in\mathbb{N}\}$ Proposed by Hans Zantema, Netherlands