Countable dense subset claim in Arzela Ascoli proof

Question

In every Arzela Ascoli proof you see the following:

Let $S = \mathbb{Q} \cap [a,b]$, where $[a,b]$ is an interval in $\mathbb{R}$, then $S$ is a countable dense subset and there exists a finite number of points $\{x_1, \ldots, x_k\}$ such that $\forall x \in [a,b], |x - x_j| < \delta$, for at least one $j = \{1, \ldots, k\}$

It seems trivially true, since $Q$ is dense in $\mathbb{R}$. But there are some interesting details....

1. there exists finitely number of points $\{x_1, \ldots, x_k\}$ , however $S$ is countable but not finite. What happened to the rest of the points?

2. For at least one $j$...? Why not for all $x_j$ since for every rational we can stick a real in between that is $\delta$ close.

Can someone provide a proof to the claim and resolve the problems?

Answer 1

Fix some $\delta > 0$. Then $\{ (q-\delta, q+\delta) \mid q \in \mathbb Q \}$ is an open covering of the compact set $[a,b]$. Therefore, there is a finite subset $\mathcal F \subseteq \{ (q-\delta, q+\delta) \mid q \in \mathbb Q \}$ that also covers $[a,b]$. This in return means that there is a finite set $\{q_1, \ldots, q_n\}$ of rational numbers such that for all $x \in [a,b]$ there is some $i \in \{1, \ldots, n\}$ with $\mid q_i - x \mid < \delta$.