Here's the problem I'm dealing with:
Let $(X,d)$ be a compact metric space and let $(U_{\lambda})_{\lambda \in \Lambda}$ be an open cover of $X$. Show that there exist $\delta >0$ such that for all $A \subset X$ with diam$A < \delta$, there exist $\lambda \in \Lambda$ such that $A \subset U_{\lambda}$.
I am not sure how to solve this one. I tried to take an open ball of radius $r= \delta + \epsilon$, for a small non-negative number $\epsilon$ which contains $A$, i.e. $A \subset B(x,r)$ for some $x \in A$. Then I consider a finite sub cover of $X$ : $(U_{\lambda})_{\lambda =1}^{n}$. Now this is also a cover of $A$, so I guess that I can find some open sets in my finite sub cover such that my ball is the union of these elements, i.e. $B(x,r)= \cup_{i=1}^k U_i$.
Is it clear that $\cup_{i=1}^k U_i$ is again contained in one of the $U_{\lambda}$ of the cover ?
Thanks for your help.