I am wondering, if a set does not contain its supremum, does this imply that the set is infinite? For example, say $A\subset\mathbb{R}$ has a supremum $a$ such that $a\notin A$. Does this imply the set is infinite? This tells us the set does not contain a maximum, but I am not sure about my other statement.
2026-04-06 13:43:53.1775483033
Set that does not contain its supremum
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You can rephrase your question as "do finite sets contain their supremum?" which is indeed true, as the supremum of a finite set is its maximum element.