Proving Bolzano-Weierstrass Theorem: A Short Guide

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Homework Help Overview

The discussion revolves around the Bolzano-Weierstrass theorem, focusing on various approaches to proving it and the dependencies on other theorems and interpretations of the theorem itself.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the length and nature of proofs, with some suggesting that the proof's complexity varies based on the theorems referenced. There is mention of needing to establish properties of sequences and subsequences as foundational steps.

Discussion Status

The conversation is ongoing, with participants sharing different perspectives on the proof's structure and the implications of various interpretations of the theorem. Some guidance on foundational concepts has been provided, but no consensus has been reached.

Contextual Notes

Participants note that the proof may differ based on the version of the theorem being considered, and there is a light-hearted exchange regarding the metaphorical aspects of the proof.

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Does anyone know the shortest way to prove Bolzano-Weierstrass theorem?
 
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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 :)
 

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