Axiom of completeness

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Caution:This is a really badly named property of the real numbers, although first years are often given it as if it were an axiom; it may be proved if one constructs the real numbers "properly"


If [ilmath]S\subseteq\mathbb{R} [/ilmath] is a non-empty set of real numbers that has an upper bound then[1]:

  • [ilmath]\text{Sup}(S)[/ilmath] (the supremum of [ilmath]S[/ilmath]) exists.


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  1. Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha