Bolzano Weierstrass Theorem Proof

Prove the Bolzano-Weierstrass theorem: Every bounded sequence has a convergent subsequence.

The Bolzano-Weierstrass theorem is a fundamental result in real analysis that underscores the interplay between boundedness and convergent subsequences within sequences. To grasp this theorem, consider a sequence that is bounded, meaning all its elements lie within a fixed range. Despite the sequence itself possibly not converging, the theorem assures that one can always extract a subsequence from it that does converge to a limit. This concept is essential because it connects to the broader idea of compactness in analysis and how it relates to limit points and convergence.

Understanding the proof of this theorem often involves constructing subsequences by leveraging the bounded property of the sequence and can be intuitively linked to the idea of partitioning the range into intervals that capture infinitely many terms. A common approach to the proof involves repeatedly bisecting the interval containing the sequence, ensuring that each bisected part contains infinitely many elements of the sequence, thereby enabling the construction of a convergent subsequence. This method highlights the close relationship between the Bolzano-Weierstrass theorem and the principle of nested intervals.

Overall, the Bolzano-Weierstrass theorem not only serves as a pillar in understanding the behavior of sequences but also provides stepping stones to exploring further concepts such as compactness in metric spaces and deepens the understanding of convergence and limit points in the real number system.