Nested Compact Set Theorem, Real Analysis I and II

This lesson proves the Nested Compact Set Property, which states that the intersection of a nested family of compact sets is non-empty. I first present three examples of nested sets, illustrating which family satisfies the theorem’s conditions. The proof is directed and uses the concept of sequential compactness, which is equivalent to compactness in a metric space. The proof involves creating a sequence where each element is chosen from a corresponding nested set. Because the first set in the nested family is compact, the sequence has a convergent subsequence whose limit lies in the first set. The key step in the proof is showing that this limit also lies in all other sets in the family. Thus, the intersection of all sets is non-empty, as the limit exists in every set in the family. The conclusion confirms that the intersection of the nested compact sets must contain at least one point. #mathematics #compactsets #topology #mathproof #metricspaces #realanalysis #theoremproof #STEM #matheducation #advancedcalculus #math #maths