Union of Real Intervals is not necessarily Real Interval
Jump to navigation
Jump to search
Theorem
Let $I_1$ and $I_2$ be real intervals.
Then $I_1 \cup I_2$ is not necessarily a real interval.
Proof
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 $I_1 \cup I_2$ is not a real interval.
$\blacksquare$
Sources
- 1964: William K. Smith: Limits and Continuity ... (previous) ... (next): $\S 2.1$: Sets: Exercise $\text{B} \ 7$