If you have a decreasing sequence of compact subsets of X such that C1⊃C2⊃... Cm and C=∩∞,m=1, then show that the Hausdorff metric, D(Cn,C)→0 as n→∞.
Assume C∈S(X)
Since C is the intersection of the s, it is sufficient to find a value of m for which every element of is within distance of C (for some given ).
Let . This is a closed (and therefore compact) subset of . It is covered by the sets , which are open in . By compactness there is a finite subcover, and since the sets form an increasing nest, there is in fact just one of them, say , that contains . It follows by taking complements that . Thus .