A text* I am reading offers a proof that a sequence of non-empty, compact, nested sets converges to its intersection in the Hausdorff metric. I do not follow the second half of the proof which shows that, in the limit, a member of the sequence is contained in the intersection in the sense that any point of that element is near some point of the intersection. The proof in the text applies the theorem that any covering of a compact set has a finite sub-cover. I do not follow the proof.
I wonder if anyone would discuss this proof with me or offer another.
*"Measure, Topology, and Fractal Geometry", by Gerald A. Edgar