Learn how upper and lower bounds lead to supremum, infimum, maximum, and minimum. This complete Real Analysis lesson includes epsilon proofs, sequence examples, the completeness axiom, and a rigorous explanation of why the rational numbers have gaps.
What does compactness mean in Real Analysis? Learn open covers, finite subcovers, the Heine–Borel Theorem, sequential compactness, Bolzano–Weierstrass, and why continuous functions become dramatically better behaved on compact sets.
![Real Analysis compactness overview showing the closed interval [0,1] and three consequences: convergent subsequences, maxima and minima for continuous functions, and uniform continuity.](https://k3p6r8v3.delivery.rocketcdn.me/wp-content/uploads/2026/08/Compactness-Slide-1.png)