I give here the example of a non-Archimedean ordered field. I know that the field is not order complete.
What is a simple example in that field of a chain without a supremum?
2026-03-26 11:15:58.1774523758
Example of a chain without a supremum in a non Archimedean ordered field
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It's the very thing that shows it's not Archimedean that witnesses that.
More specifically, $\Bbb N$, is such chain. If $x$ is the supremum of all the natural numbers, then $x-1$ should be smaller than a natural number $k$ and therefore $x=(x-1)+1$ should be at most $k+1$ which is a contradiction.