Prove every bounded sequence has a convergent subsequence (The Bolzano-Weierstrass Theorem)

This video references the book "Introduction to Real Analysis" by Bartle and Sherbert (Fourth Edition). The fact we are proving in this video is called the Bolzano-Weierstrass Theorem, and it is given as Theorem 3.4.8 in the book. See Section 3.4 for more details. Here are the two preliminary results we use in the proof: Prop (1):    • Prove that every sequence has a monotone s...   Prop (2):    • Proof of the Monotone Convergence Theorem ...   Thanks and enjoy the video! Real Analysis (Bartle and Sherbert) Playlist:    • Real Analysis [ILIEKMATHPHYSICS]  

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