Let $X$ be a compact metric space, and $\{U_{\lambda}:\lambda\in\Lambda\}$ an open cover of $X$. Let $\delta$ be its Lebesgue number. I want to show that the sets $F_\lambda=\{x\in U_\lambda:d(x,\partial U_\lambda)\leq \delta\}$ also cover $X$.
As a start, $x\in F_\lambda$ means that $x\in B(y,\delta) \cap U_{\lambda}$ for $y \in \partial U_{\lambda}$. I do not see why $F_\lambda$ cover $X$.
Thank you for any help.
:: I just noticed that as in the proof of the Lebesgue Number lemma, $d(x,\partial U_\lambda):=\inf_{y \in \partial U_\lambda}d(x,y)$.