In baby Rudin:
1.10 Definition:
An ordered set S is said to have the least-upper-bound property if the following is true:
If E $\subset$ S, E is not empty, and E is bounded above, then sup E exists in S.
I find the way it's written to be weird, shouldn't he instead have written:
If E $\subset$ S, E is not empty, and E is bounded above, and sup E exists in S.
I mean it's a definition, he can't come to conclusions in a definition.. Please someone explain I'm really confused thanks a lot!!
Best Answer
It should actually be as follows:
An ordered set S is said to have the least-upper-bound property if and only if the following is true:
For all E, $\;\;\;$ if $\;$ $\:$$\{\hspace{-0.02 in}\}$ $\neq$ E $\subset$ S$\:$ and E is bounded above $\:\:$ then $\:\:$ sup E exists in S $\;\;\;$.
.