Finding the supremum and infimum of the following set Let $A = (2,5) \cup \{7\}$. Find the supremum and infimum of $A$.
Firstly, I claim that $\sup A = 7$ and $\inf A = 2$. But, I got confused that in $A$, there is a singleton set. How to approach the supremum? And also, the infimum?
An approach that I know is using this theorem:
Let $S \subseteq \Bbb R$ and suppose that $s:= \sup S$ belongs to $S$. If $u \notin S$, then
$\sup(S \cup \{u\}) = \sup \{s,u\}$.
Thanks in advanced.

A sequence that converges to the supremum is $7, 7, \ldots$.
$7$ can be taken since it is an element of $A$.

Edit:
$7$ is an upper bound of $A$. Furthermore, any upper bound of $A$ must be bigger than $7$, hence $7$ is the least upper bound.
Similarly for the lowerbound.
