Prove that $\cup_{\alpha} A_{\alpha} $ has finite diameter if $\cap_{\alpha} A_{\alpha}≠∅$ and there exists a constant $M$ such that $diam(A_{\alpha}) ≤ M$ for all $\alpha$. Each $A$ is a subset of metric space $(X,d)$, and each $A$ has a finite diameter (that is, $diam(A) = sup{d(x,y): x,y ∈A}$).
Not too sure where to begin, any help appreciated. Note ∝ is a subscript.
Pick a point in the intersection. Then every point in your union is at distance at most what of that?