Bolzano-Weierstrass property

The Bolzano-Weierstrass property, also known as the Weierstrass property, is a fundamental concept in mathematical analysis that deals with the convergence of sequences. It states that every bounded infinite sequence of real numbers has at least one convergent subsequence. This property is crucial for proving many other theorems in analysis, such as the intermediate value theorem and the extreme value theorem.12

The formal definition of a limit, developed by Karl Weierstrass, is key to understanding this property. It precisely defines how a sequence approaches a limit, ensuring that terms in the sequence can be made arbitrarily close to the limit by choosing a sufficiently large index. This rigorous foundation, established through the arithmetization of analysis by mathematicians like Bolzano, Dedekind, and Weierstrass, allows for the precise treatment of concepts like continuity and convergence, which are essential for the Bolzano-Weierstrass property.12