Suppose I have an ordered set $A$ and a chain $B\subseteq A$ then does $B$ necessarily have a supremum? Let alone an upper bound? And if it is empty? This question is a bit confusing because I am not sure if I am expected to prove a classic case or to notice little cases in which the claim isn't true.
2026-04-04 02:44:53.1775270693
Ordered sets. Chain upper bounds.
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Consider $\mathbb{N}$ with its usual ordering. Does the chain $\mathbb{N}\subseteq\mathbb{N}$ have a supremum?