In Rudin's book the following are defined as:
Suppose $S$ is an ordered set, $E$ is a proper subset of $S$, and $E$ is bounded above. Suppose there exists an $\alpha$ in $S$ with following properties:
a) $\alpha$ is an upper bound of $E$
b) If $\gamma < \alpha$ then $\gamma$ is not an upper bound of $E$.
Then $\alpha = \sup E$
Least Upper Bound Property
An ordered set $S$ is said to have the LUBP if the following is true:
If $E$ is a proper subset of $S$, $E$ is not empty, and $E$ is bounded above, then $\sup E$ exists in $S$.
Doesn't the first definition imply that if $\sup E$ exists, then $\sup E$ is necessarily in $S$, from the assumption the $\alpha$ is in $S$ ?
And so shouldn't ending of the definition of the LUBP be changed to "$\sup E$ exists.", rather than, "$\sup E$ exists in $S$" ?