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