When does an ordered set have a supremum?

Question

My textbook states that an ordered set S has the least upper bound property if the following is true:
If $E \subset S, E $ is not empty and is bounded above, then $sup E$ exists in S.
But consider the set of rational numbers $\mathbb Q$, and a subset $S = {{1,2,3}}$. $S$ is not empty. It's also bounded above. $sup S=3$ also exists in $\mathbb Q$. But the set of rational numbers does not have a supremum!

Answer 1

Not, The set $S$ has a supremum but it is in $\mathbb{R}$. The supremum (and infimum) of a set can be not in that set.

Answer 2

It is possible that the sum and inf are not in the set. For example; consider the set

$$E= \{ \frac{1}{x}| x \in \mathbb{N}\}.$$

It is clear that $\inf{E}=0$ but $0 \not \in E$.