The finite covering theorem

This animated video demonstrates the proof of the Heine-Borel Theorem (Finite Covering Theorem). It begins by assuming for contradiction that a closed interval [a, b] has an open cover with no finite subcover. Through recursive bisection, it constructs a nested sequence of closed intervals, none of which can be finitely covered. Applying the Nested Interval Property, a unique point ξ is found at the intersection. Since the cover consists of open sets, one set must contain a neighborhood of ξ, eventually covering an entire small interval from the sequence. This contradicts the initial assumption, proving that every open cover of a compact interval admits a finite subcover.

Before you watch

• Real Analysis Basics
• Definition of Open Sets and Covers
• Completeness Axioms of Real Numbers