Compact Subsets Are Closed and Bounded

Prove that if a subset of $\R^P$ is compact, then it is closed and bounded.

In real analysis, one of the essential concepts learned is compactness, which provides a bridge between finite and infinite dimensional spaces by extending the notion of a set being both limited in extent and encompassing all of its boundary points. Compactness serves as a generalization of closed and bounded sets, ensuring some key properties that facilitate deeper results in analysis, such as the Bolzano-Weierstrass theorem.

This problem explores the concept that any compact subset of a Euclidean space, like R to the power of P, must be closed and bounded. To approach this problem, you should recall the Heine-Borel theorem, which characterizes compact sets in Euclidean spaces, asserting that a set is compact if and only if it is closed and bounded. Thus, this problem asks you to establish this bidirectional relationship by demonstrating that compactness implies closedness and boundedness.

While proving this statement, consider the properties of compactness such as every open cover having a finite subcover. This property can be used to show that limits of convergent sequences within the set remain within the set (closedness) and that the entire set can be contained within some large ball in the space (boundedness). Understanding the nature of compact sets is pivotal for tackling many problems in analysis involving continuity, functional limits, and optimization on real-valued functions.