Theorem

Bolzano-Weierstrass Theorem

Analysis

The Bolzano-Weierstrass Theorem states that every bounded sequence of real numbers has a convergent subsequence. Named for Bernard Bolzano and Karl Weierstrass, it is a foundational compactness result in real analysis and underlies proofs of the Heine-Borel Theorem and the existence of extrema of continuous functions on closed bounded intervals.

Facts

Statement

Every bounded sequence in a finite dimensional Euclidean space has a convergent subsequence. 1

Proof Year

1817 1

Classification

Statement Form

Existence Theorem 1

Connections

Associated With

Heine-Borel Theorem

Has Statement Form

Existence Theorem

Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.

In Branch

Analysis

Proved By

Bernard Bolzano

Karl Weierstrass

Sources

1. Bolzano-Weierstrass Theorem (Wikipedia)

Wikimedia Foundation

History and significance section
It was actually first proved by Bolzano in 1817 as a lemma in the proof of the intermediate value theorem.
Lead section
The theorem is sometimes called the sequential compactness theorem.

Switch Atlas

The Inner World

The Civil World

The Natural World

Bolzano-Weierstrass Theorem

Associated With

Has Statement Form

In Branch

Proved By