disconnected ordered set

Question

Is there a totaly ordered infinite set $A$ with the least element $a$ and the greatest element $b$ such that for any sequence $\{\alpha_n\}$ and $\{\beta_n\}$ in $A$ which satisfies $\alpha_n<\beta_n$, we can deduced that $\lim\alpha_n<\lim\beta_n$ (or even $\sup \alpha_n<\sup\beta_n$) ?( the last inequalities is remained strict)

Answer 1

If $a_n$ is any sequence with $a':=\lim \alpha_n$ such that $\alpha_n< a'$ for all $n$, then set $\beta_n:=a'$ for a counterexample.

Answer 2

The total order $ \omega + \omega^*$ (that is, the order type of the natural numbers followed by the reverse of the order type of the natural numbers) is an example of such a total order. The only sequences in this order that have a limit are eventually constant.