Set Union/Examples/Set of Unbounded Above Open Real Intervals

Example of Union of Set of Sets
Let $\R$ denote the set of real numbers.

For a given $a \in \R$, let $S_a$ denote the (real) interval:
 * $S_a = \openint a \to = \set {x \in \R: x > a}$

Let $\SS$ denote the family of sets indexed by $\R$:
 * $\SS := \family {S_a}_{a \mathop \in \R}$

Then:
 * $\displaystyle \bigcup \SS = \R$.

Proof
By construction we have that $\SS \subseteq \R$.

It remains to prove that $\R \subseteq \SS$

$x \in \R$ such that $x \notin \SS$.

Then by definition of set union:
 * $\not \exists a \in \R: x \in S_a$

That is:
 * $\forall a \in \R: x \le a$

But then we have:
 * $x - 1 \in \R$

leading to:
 * $x < x - 1$

which contradicts the properties of real numbers, which lead to:
 * $x > x - 1$

Hence the result.