Proving given sets are mathematical intervals

Question

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?

Hint. $S\cup G\subseteq [a,b]$ for $a\in L$ and $b\in U$. – Hanul Jeon Oct 23 '19 at 07:29 — Hanul Jeon, Oct 23 '19 at 07:29

score 1 · Accepted Answer · answered Oct 23 '19 at 07:31

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 $c \in S \cup G.$

Proving given sets are mathematical intervals

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