Proving Bolzano-Weierstrass with Nested Interval Property | Real Analysis

We prove the Bolzano Weierstrass theorem using the Nested Interval Property. The Bolzano-Weierstrass theorem states every bounded sequence has a convergent subsequence. We will construct a subsequence by bounding our sequence between M and -M, then creating an infinite sequence of nested intervals from which the terms of our subsequence will come. We prove this subsequence converges to prove the BW theorem. #realanalysis

Proof of Nested Interval Property:    • Nested Interval Property and Proof | ...  
Proof of Bolzano Weierstrass Theorem for Sequences (with Monotone Convergence Theorem):    • Short Proof of Bolzano-Weierstrass Th...  
Proof that x^n Converges when |x| is less than 1: (coming soon)

Credit to Understanding Analysis by Stephen Abbott, where I first read this proof, and whose model I followed for the presentation.

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