Theorem

Let $\mathcal{A}$ be an Open cover of the compact, metric sppace $X$. Then there exists a $\delta > 0$ (a Lebesgue number associated to the open covering $\mathcal{A}$) such that:

every subset of $X$ with a diameter $<\delta$ is contained in at least one $A \in \mathcal{A}$

Proof. It is enough to assume Sequentially compact actually.