The Bolzano-Weierstrass theorem is a fundamental result in real analysis that states every bounded sequence in \mathbb{R} has a convergent subsequence. This means that if you have a sequence of real numbers that does not go off to infinity, you can always find a smaller sequence that approaches a specific limit. This theorem is particularly important in the study of compactness and continuity in mathematics. It helps establish the behavior of functions and sequences, making it a key concept in various fields, including calculus and optimization.