Difference of Unbounded Closed Intervals

Theorem
Let $a, b \in \R$ have $a < b$.

Then:


 * $\hointl {-\infty} b \setminus \hointl {-\infty} a = \hointl a b$

where $\setminus$ denotes set difference.

Proof
Note that:


 * $x \in \hointl {-\infty} b \setminus \hointl {-\infty} a$




 * $x \in \hointl {-\infty} b$ but $x \not \in \hointl {-\infty} a$.

That is:


 * $x \le b$ but it is not the case that $x \le a$.

So this is equivalent to:


 * $x \le b$ and $x > a$.

That is:


 * $x \in \hointl a b$

So:


 * $\hointl {-\infty} b \setminus \hointl {-\infty} a = \hointl a b$