Bolzano-Weierstrass Theorem

The Bolzano-Weierstrass Theorem is a fundamental result in real analysis that states every bounded sequence of real numbers has a convergent subsequence. This theorem plays a crucial role in the study of limits, continuity, and differentiability in calculus.

More formally, if a sequence of real numbers {a_n} is bounded, then there exists a subsequence {a_{n_k}} that converges to a limit L.

The Bolzano-Weierstrass Theorem is significant because it guarantees the existence of convergent subsequences for bounded sequences, even if the original sequence does not converge. This theorem is used in various proofs and theorems in calculus, including the Compactness Theorem and the Extreme Value Theorem.