When does the Least upper bound property fail

Question

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.

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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.

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edit - adding definition from text

This is the definition from the text that I'm using :

Answer 1

Let $S$ be a set equipped with a linear order $<$.

Then order $\langle S,<\rangle$ has by definition the least upper bound property if every non-empty subset $E$ that is bounded above has a least upper bound (which is called the supremum of $E$).

This for instance is the case in $\langle\mathbb R,<\rangle$.

But it is not the case in e.g. $\langle\mathbb Q,<\rangle$.

Take subset $E=\{q\in\mathbb Q\mid q<\pi\}$. Then $E$ is not empty and bounded above (any $b\in\mathbb Q$ with $\pi<b$ serves as upper bound of $E$) but there is no least upper bound (note that $\pi\notin\mathbb Q$).

Answer 2

"What this seems to be saying to me", namely that if I take something $E$ less than or equal to something $S$, then the largest element in $E$ will be in $S$ does not describe the least upper bound property of $S$.

The least upper bound property of $S$ says the following: If $E$ is a nonempty subset of $S$, and the set ${\rm upb}(E)\subset S$ of all upper bounds of $E$ is nonempty then ${\rm upb}(E)$ has a least element $\sigma$. This $\sigma$ is uniquely determined, and is denoted by $\sup E$. Sometimes $\sup E\in E$, in which case $\sup E$ is called the maximal element of $E$, and is denoted by $\max E$.