Question: Let $\{O_{\lambda}:\lambda\in\Lambda\}$ be an open cover of the metric space $(X,d).$ If $X',$ the set of all accumulation points of $X$ is compact then show that $\exists~\delta>0$ such that for $x\in X'$ there exists $\lambda_x\in\Lambda$ such that $B(x,\delta)\subset O_{\lambda_x}.$
Unfortunately I could not solve it.
Please help.
My try: I tried to think of a solution using the following result: If $A$ is a compact subset of a metric space $(X,d)$ then for any open set $U$ containing $A$, there exists a $\delta>0$ such that $B(A,\delta) \subset U.$