Infimum and Supremum the same

Question

Given a set A in R,

Can the infimum of A be the same as the supremum of A? If so, does that mean the set A only have one element?

Answer 1

Yes, one point sets have the same supremum and infimum (actually the same maximum and minimum). If a set $A$ has more two different elements $x<y$ then $\inf A\le x<y\le \sup A$ so their supremum and infimum (in case of existing) are different.

Answer 2

Yes. For any $a \in A$ $\inf A \leq a$, $\sup A \geq a$, since these are the same $\inf A = \sup A = b$ means $b\geq a$ and $a \geq b$, that is $a=b$.

Answer 3

A single element set has the same supremum and infimum. If a set has two distinct real numbers (or more) then we can pick two different elements $a$ and $b$. Let $b$ be the larger of the two. Then the supremum is bigger than $b$ and infimum smaller than $a$, and hence the two are different.