Definition
Bolzano–Weierstrass Theorem
The Bolzano–Weierstrass theorem states that every bounded sequence in has at least one limit point.
Equivalently, every bounded sequence in contains a convergent subsequence.
The theorem is named after Bernard Bolzano and Karl Weierstrass. It is a fundamental result in analysis that characterises compactness in Euclidean space.
Significance
The Bolzano–Weierstrass theorem is crucial because it guarantees that we can always find a “limit” (in the form of a subsequence limit) for any sequence that does not escape to infinity. It is used to prove several other key theorems, such as the Extreme Value Theorem and the completeness of the real numbers.