Union of Real Intervals is not necessarily Real Interval

Theorem
Let $I_1$ and $I_2$ be real intervals.

Then $I_1 \cup I_2$ is not necessarily a real interval.

Proof
Proof by Counterexample:

Consider the real intervals:
 * $I_1 = \left({0 \,.\,.\, 2}\right)$
 * $I_2 = \left({4 \,.\,.\, 6}\right)$

Then we have that:
 * $1 < 3 < 5$

where:
 * $1 \in I_1 \cup I_2$
 * $5 \in I_1 \cup I_2$

but:
 * $3 \notin I_1 \cup I_2$

Thus by Interval Defined by Betweenness, $I_1 \cup I_2$ is not a real interval.