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).

Proof sketch (optional): If no such $\delta$ exists, choose points $x_n$ with balls $B(x_n,1/n)$ not contained in any single cover set. By compactness, extract a convergent subsequence $x_{n_k}\to x$. Since $x$ lies in some $U\in\mathcal{U}$ and $U$ is open , a small ball around $x$ lies in $U$, forcing large $k$ to have $B(x_{n_k},1/n_k)\subseteq U$, a contradiction.