Lebesgue Number Lemma

Lebesgue Number Lemma: Let $(X,d)$ be a compact metric space and let $\mathcal{U}$ be an open cover of $X$. Then there exists $\delta>0$ (a Lebesgue number for $\mathcal{U}$) such that for every $x\in X$, $B(x,\delta)\subseteq U \quad \text{for some } U\in\mathcal{U}.$

This lemma is used to pass from pointwise local control to uniform control on compact sets (e.g., in proofs of uniform continuity and partitions of unity in more advanced settings).