I'm currently having some trouble proving/disproving the existence of what’s stated in the title. Everywhere I've looked, people say that this is impossible, and others say that it $\textit{is}$ possible. I say that it is possible, and this is my logic (or lack thereof):
By the Axiom of Completeness, every nonempty set of reals that's bounded above has a supremum. However, we know that a set $A$'s supremum need not be an element of this set. We also know that a real number $a_0 \geq a$ for all $a \in A$ is a maximum if it is in the set $A$. If we let $A = \{x \in \mathbb{R} : 0 \leq x < 2\}$, then we see that this set does not possess a maximum. Its supremum exists but is not $\textit{contained}$ in the set, but the infimum is, as desired.
What I see others say online is, "This is impossible because for any finite set, its supremum equals its maximum, which is always contained in the set." But, isn't it the case that if a maximum exists, then it's the supremum, and that the supremum can exist without it being the maximum? Any help is appreciated.
Best Answer
The set $A$ you provided is bounded, but it's not finite.
If we have a non-empty finite set, then we can sort its elements in increasing order, say $x_1\leq x_2\leq x_3\leq ... \leq x_n$, now notice that $x_1$ is the smallest element in our set, as such it's the infimum and minimum (similarly for $x_n$ is the supremum and maximum).
Take away: If a set is finite, then it has an infimum which is the minimum of that set, and a supremum which is the maximum of that set.
Now for infinite sets, the minimum, maximum, supremum and infimum may exist $[1,2]$. We can also find a set with infimum, supremum and maximum $(1,2]$, a set with infimum, minimum and supremum $[1,2)$, a set with infimum and supremum only $(1,2)$, a set with infimum and minimum only $\mathbb{N}$ which may be $0$ depending on the axioms of your choice, infimum only like $\mathbb{R}_{>0}$, supremum and maximum like $\mathbb{Z}_{<0}$ which is $-1$, supremum only like the set $\mathbb{R}_{<0}$, or a set with no infimum and supremum like $\mathbb{R}$.