Is there any ordered field S with a maximum value?
i.e.,exist $n\in S$ such that $\forall x\in S$ \ {$n$},$x\le n$. What if '$\le$' is changed by '$\lt$'?
In the book Elementary Analysis: The theory of Calculus (Kenneth A. Ross),Edition 2, P.24, it says:
In other words, there exists a mathematical system satisfying all the properties A1– A4, M1–M4, DL and O1–O5 in §3 and yet possessing elements $a > 0$ and $b > 0$ such that $a < 1/n$ and $n < b$ for all $n$. <
As mentioned in comment by Henry, does hyperreal number or surreal number help?
I wonder an example.
Thanks