Let $\mathbb{Z}^{\mathbb{N}}$ to be the set of all sequences of integers. Where every element $x$ is of the following form: $x=(x_{1},x_{2},...)$ with $\forall n\in \mathbb N, x_{n}\in\mathbb{Z}$.
We order the set as follows: $x\lneq y\Leftrightarrow$ the first non-zero entry of $y-z$ is positive. Where we denote subtractions as term by term such that: $y-z$ corresponds to the sequence $(y_{1}-x_{1},y_{2}-x_{2},...,)$
Assume that $(\mathbb Z^\mathbb N, \lneq)$ forms a totally ordered set.
I’m trying to find an example of some element in this totally ordered set that has an upper bound but does not have a supremum.
So far I’ve been unable to come up with one, any advice is appreciated!