Prove that Corollary 3.13, for real sequences, is equivalent to the Least Upper

Question

Prove that Corollary 3.13, for real sequences, is equivalent to the Least Upper Bound Property. More precisely, assume all the axioms for the reals from Section 2A except O6, and assume that Corollary 3.13 is valid for real sequences. Show that O6 is a consequence of these assumptions. Corollary 3.13: The Bolzano-Weierstrass Theorem. Each bounded real or complex sequence has a convergent subsequence. O6 If A is any nonempty subset of IR that is bounded above, then there is a least upper bound for A.

Explanation / Answer

your complete answer is at [PDF] CONSTRUCTION OF R www.iitg.ernet.in/kvsrikanth/teaching/2011f/reals.pdf

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