I have the least upper bound property as follows
For a subset $E$ of $S$, if $S$ has the least upper bound property then the supremum of $E$ is in $S$
What this seems to be saying to me, is that if I take something $E$ less than or equal to something $S$, then the largest element in $E$ will be in $S$.
Which seems quite obvious, which makes me think I'm not appreciating what's really being said here.
Perhaps some counter examples would help me to fully appreciate this property.
I reviewed some of the other questions.
Proving that ℝ satisfies the Least Upper Bound property discussed metric spaces and proving the property.
Is Wikipedia wrong about the least-upper-bound property? spoke about something specific relating to wikipedia.
edit - adding definition from text
This is the definition from the text that I'm using :