Lebesgue number lemma refinement lemma

Refinement lemma (used for Lebesgue numbers): Let $(X,d)$ be a metric space, let $K\subseteq X$ be compact, and let $\mathcal{U}$ be an open cover of $K$. For each $x\in K$, choose $U_x\in\mathcal{U}$ with $x\in U_x$. Since $U_x$ is open, there exists $r_x>0$ such that

Then there exist points $x_1,\dots,x_N\in K$ such that

In particular, if $\delta=\min_{1\le i\le N} \frac{r_{x_i}}{2}$, then $\delta>0$ and for every $x\in K$ there exists $U\in\mathcal{U}$ with $B(x,\delta)\subseteq U$.

This is the standard compactness step that produces a uniform scale from pointwise local containment. See also Lebesgue number lemma.