Teorema de Heine-Borel

The Heine-Borel Theorem states that in a finite-dimensional space like the real line or Euclidean space, a set is "compact" if and only if it is both closed and bounded. "Closed" means it contains all its limit points, and "bounded" means it fits within some finite region. This is important because compact sets have properties like ensuring that continuous functions reach their maximum and minimum values within them. Essentially, the theorem links geometric size and boundary behavior to a key mathematical property that guarantees certain desirable features in analysis and calculus.