The Bolzano-Weierstrass theorem states that every bounded sequence of real numbers has a convergent subsequence. This means that the set of real numbers equ...
The Bolzano-Weierstrass theorem states that every bounded sequence of real numbers has a convergent subsequence. This means that the set of real numbers equipped with the Hausdorff metric is complete with respect to the topology of sequential convergence.
In other words, every sequence of real numbers has a limit point. A limit point is a point that every neighborhood of the point contains infinitely many points in the sequence.
This theorem is a powerful generalization of the completeness of the real number line. The Bolzano-Weierstrass theorem implies that the real number line is complete with respect to the Hausdorff metric, which is the natural metric induced by the distance function.
Here's an example to illustrate the Bolzano-Weierstrass theorem:
Consider the sequence of real numbers (0, 1, 2, 3, 4, 5). This sequence is bounded, but it does not converge to any real number. However, the Bolzano-Weierstrass theorem guarantees that the sequence has a convergent subsequence, such as the sequence ([0, 1/2, 1/4, 1/8, 1/16])