Open Real Interval is Subset of Closed Real Interval
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Theorem
Let $a, b \in \R$ be real numbers.
Then:
- $\openint a b \subseteq \closedint a b$
where:
- $\openint a b$ is the open interval between $a$ and $b$
- $\closedint a b$ is the closed interval between $a$ and $b$.
Proof
Let $x \in \openint a b$.
Then by definition of open interval:
- $a < x < b$
Thus:
- $a \le x \le b$
and so by definition of closed interval:
- $x \in \closedint a b$
$\blacksquare$
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 2$: Sets and Subsets