In my mind the axiom of choice is about being able to choose one element from each subset of some universal set $X$. I dont see the connection to the Axiom of choice in the part of the proof below. 

Let $u(x): \mathbb{R}^2 \rightarrow \mathbb{R}$ with $u(x,2) > u(x,1) \; \forall \; x \in \mathbb{R}$. By density of $\mathbb{Q}$ in $\mathbb{R}$, $ \exists \, q(x) \in \mathbb{Q} : u(x,2) > q(x) > u(x,1).$ 

**Having done that for every $x \in \mathbb{R}^+$, we obtain a function $q: \mathbb{R}^+ \rightarrow \mathbb{Q}$ by the Axiom of choice.**