Proving Bolzano-Weierstrass Theorem: A Short Guide

• glebovg
In summary, 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. It is named after mathematicians Bernard Bolzano and Karl Weierstrass, who independently proved it in the 19th century. The theorem is significant because it provides a powerful tool for proving the convergence of sequences and the existence of limits in real analysis, and is a key result in the proof of the Heine-Borel theorem. The Bolzano-Weierstrass theorem holds for a sequence of real numbers if and only if the sequence is bounded and infinite. Variants of the theorem
glebovg
Does anyone know the shortest way to prove Bolzano-Weierstrass theorem?

The length of your proof will depend on what theorems you're allowing yourself to call upon. I recall having to show first that all sequences have monotonic subsequences, and that any bounded monotonic subsequence must converge. Bolzano-Weierstrass is a consequence of that.

it also depends on which version of the theorem you're proving.

i always liked the: "the lion lives somewhere in the jungle" proof.

I always thought lions lived on the savannah...

this is true, but it doesn't affect the proof :)

What is the 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.

Who discovered the Bolzano-Weierstrass theorem?

The theorem is named after mathematicians Bernard Bolzano and Karl Weierstrass, who independently proved it in the 19th century.

What is the significance of the Bolzano-Weierstrass theorem?

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.

What are the conditions for the Bolzano-Weierstrass theorem to hold?

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.

Can the Bolzano-Weierstrass theorem be extended to other spaces?

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.

• Calculus and Beyond Homework Help
Replies
14
Views
1K
• Calculus and Beyond Homework Help
Replies
12
Views
1K
• Calculus and Beyond Homework Help
Replies
8
Views
2K
• Calculus and Beyond Homework Help
Replies
9
Views
5K
• Calculus and Beyond Homework Help
Replies
6
Views
884
• Calculus and Beyond Homework Help
Replies
4
Views
1K
• Topology and Analysis
Replies
2
Views
903
• Calculus and Beyond Homework Help
Replies
2
Views
2K
• Calculus and Beyond Homework Help
Replies
3
Views
628
• Calculus and Beyond Homework Help
Replies
9
Views
405