Does any totally ordered set $S$ does not have a maximal element have a sequence $\{x_n\}_{n \in \mathbb{N}}$ of elements in the set such that the sequence is unbounded?
In your answer I am looking for a proof, counterexample, or reference.
2026-03-25 04:37:46.1774413466
Unbounded sequences in a general totally ordered set without max
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No, consider the totally ordered set $[0,1]$.
Update:
Still. No. Consider the totally ordered set $\omega_1$ of countable ordinals. Since the sup of a countable sequence of countable ordinals, is again a countable ordinal, every countable sequence is bounded in $\omega_1$. Moreover, as $\omega_1$ is a limit ordinal, it has no largest element.