I don't understand the proof of the Proposition 2.3.4. in the book "Optimal Control" by Richard Vinter.
Proposition 2.3.4.: Consider a function $\phi\colon I\times \mathbb{R}^{n} \times\mathbb{R}^{m}\to \mathbb{R}^{k}$ satisfying the followinghypotheses.
- $\phi(t,\cdot,u)$ is continuous for each $(t,u)\in I\times\mathbb{R}^{m}$.
- $\phi(\cdot,x,\cdot)$ is $\mathcal{L}\times\mathcal{B}^{m}$-measurable for each$x\in\mathbb{R}^{n}$. Then, for any Lebesgue measurable function$x\colon I\to\mathbb{R}^{n}$, the mapping $(t,u)\to \phi(t,x(t),u)$ is$\mathcal{L}\times\mathcal{B}^{m}$-measurable.
At the begin of the proof, the author consider $\{r_{j}\}$ an ordering of the set of $n$-vectors with rational coefficients. For each $k\in\mathbb{N}$ define:$$\phi_{k}(t,u) := \phi(t,r_{j},u),$$where $j$ is chosen such that:$$|x(t)-r_{j}| \leq \frac{1}{k}$$and$$|x(t)-r_{i}| > \frac{1}{k}$$for all $i=1,2,\dots,j-1$. These conditions uniquely define $j$ (note that $j$ depend of $t$ and $k$), but I don't understand why. Perhaps it is an argument typical of the real analysis that I cannot see or do not remember, I would appreciate if someone helps me understand that step.
Regards