Let’s assume we have a subset of real numbers called $S$. We call the set of upper bounds of $S$, $U$ and we call its set of lower bounds, $L$.
If we define a set called $G$ that consists of any real number not included in the sets $S$, $U$ and $L$, how could we prove that $U$, $L$ and especially, $S \cup G$ are all mathematical intervals?
You use the definition. That is, to prove $X$ is an interval, suppose $a,b \in X$ and $a < c < b.$ Then show $c \in X.$
For $L,$ show that $c < b$ and $b \in L$ implies $c \in L.$
$U$ is similar to $L.$
For $S \cup G,$ show that $c < b$ and $b \in S \cup G$ implies $c \not\in U.$ Similarly, $a < c$ and $a \in S \cup G$ implies $c \not\in L.$ Conclude $c \in S \cup G.$