Question 3
Let $\mathbb{Q}_{\gt 0}$ be the set of positive rational numbers. Let $f: \mathbb{Q}_{\gt 0} \rightarrow \mathbb{R}$ be a function satisfying the conditions
$$
\begin{aligned}
& f(x) f(y) \geqslant f(x y) \\
& f(x+y) \geqslant f(x)+f(y)
\end{aligned}
$$
for all $x, y \in \mathbb{Q}_{\gt 0}$. Given that $f(a)=a$ for some rational $a\gt 1$, prove that $f(x)=x$ for all $x \in \mathbb{Q}_{\gt 0}$.
$$
\begin{aligned}
& f(x) f(y) \geqslant f(x y) \\
& f(x+y) \geqslant f(x)+f(y)
\end{aligned}
$$
for all $x, y \in \mathbb{Q}_{\gt 0}$. Given that $f(a)=a$ for some rational $a\gt 1$, prove that $f(x)=x$ for all $x \in \mathbb{Q}_{\gt 0}$.