Open Unit Interval is Proper Subset of Closed Unit Interval

Theorem

The open unit interval:

$I_o = \openint 0 1$

is a proper subset of the closed unit interval:

$I_c = \closedint 0 1$

Proof

Let $x \in I_o$.

Then by definition:

$0 < x < 1$

and so:

$0 \le x \le 1$

and so:

$x \in I_c$.

Thus:

$I_o \subseteq I_c$

Consider:

$0 \in I_c$

by definition of closed interval.

But it is not the case that $0 < 0$.

So $0 \notin I_o$ and so $I_c \nsubseteq I_o$.

Hence the result by definition of proper subset.

$\blacksquare$

Sources

• 1964: William K. Smith: Limits and Continuity ... (previous) ... (next): $\S 2.1$: Sets