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glebovg
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Does anyone know the shortest way to prove Bolzano-Weierstrass theorem?
The Bolzano-Weierstrass theorem, also known as the Bolzano-Cauchy theorem, is a fundamental theorem in real analysis that states that every bounded sequence of real numbers has a convergent subsequence.
The theorem is named after mathematicians Bernard Bolzano and Karl Weierstrass, who independently proved it in the 19th century.
The Bolzano-Weierstrass theorem is significant because it provides a powerful tool for proving the convergence of sequences and the existence of limits in real analysis. It is also a key result in the proof of the Heine-Borel theorem, which characterizes compact sets in Euclidean space.
The Bolzano-Weierstrass theorem holds for a sequence of real numbers if and only if the sequence is bounded, meaning that its terms do not become infinitely large, and it is infinite, meaning that it has an infinite number of terms.
Yes, variants of the Bolzano-Weierstrass theorem have been developed for other spaces, including Banach spaces, Hilbert spaces, and metric spaces. These versions typically require additional conditions, such as completeness or compactness, in order to hold.