Question 5
Suppose a function $f \colon \mathbb{Q}_{\gt 0} \to \mathbb{R}$ satisfies:
1. [(i)] If $x,y \in \mathbb{Q}_{\gt 0}$, then $f(x)f(y) \ge f(xy)$.
2. [(ii)] If $x,y \in \mathbb{Q}_{\gt 0}$, then $f(x+y) \ge f(x) + f(y)$.
3. [(iii)] There exists a rational number $a \gt 1$ with $f(a) = a$.
Prove that $f(x) = x$ for all positive rational numbers $x$.
1. [(i)] If $x,y \in \mathbb{Q}_{\gt 0}$, then $f(x)f(y) \ge f(xy)$.
2. [(ii)] If $x,y \in \mathbb{Q}_{\gt 0}$, then $f(x+y) \ge f(x) + f(y)$.
3. [(iii)] There exists a rational number $a \gt 1$ with $f(a) = a$.
Prove that $f(x) = x$ for all positive rational numbers $x$.