Is supremum part of the set or it is the bigest element out of it?

Question

"If $\sup A$ is in $A$, then is $\sup A$ called also Maximum: $\max A$."

So that means that $\sup A$ can be outside the set $A$? And lastly the upper barrier(or bound, not sure) is from $A$, right?

Answer 1

You can have sets that don't contain their supremum. A simple example is the set $(0,1)$: the supremum of this set is 1 since 1 is greater than or equal to any element of this set, but it is also the lowest possible upper bound. Clearly 1 is not in the set either.

To your second question, if I'm reading it right, one can similarly say that an upper bound is not necessarily in the set, and you can even say it won't be in the set if it is not the supremum..

Answer 2

http://en.wikipedia.org/wiki/Infimum_and_supremum would note that the supremum can be outside the set as various examples demonstrate:

sup { x ∈ ℚ: $x^2$ < 2 } = $\sqrt2$

The upper bound may or may not be part of the set. In the example above, the bound isn't part of the set.

Answer 3

Yes, that what it means. For example, $\sup (0,1)=1\notin (0,1)$. And since $\sup A$ is the least upper bound, if $\sup A\notin A$ it means that there is no element of $A$ which is an upper bound of $A$.