Proving Bolzano-Weierstrass Theorem: A Short Guide

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The discussion centers on the Bolzano-Weierstrass theorem and the most efficient methods to prove it. A key point is that the proof's length varies based on the theorems used as prerequisites. It is noted that demonstrating the existence of monotonic subsequences is essential, as any bounded monotonic subsequence must converge, leading to the theorem's conclusion. Additionally, the version of the theorem being proved influences the approach taken. Overall, the conversation highlights different proof strategies while emphasizing the theorem's foundational concepts.
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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 :)
 

Thread 'Distance between a Clock's hands when the distance is increasing most rapidly'

Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?
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