Set Union/Examples/Finite Subfamily 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 $\R$:
 * $\SS := \family {S_a}_{a \mathop \in \R}$

Let $\TT$ be a finite subfamily of $\SS$.

Then:
 * $\ds \bigcup \TT$ is a proper subset of $\R$.

Proof
According to the definition of subfamily, let $\TT$ be the family of sets indexed by $U$, where $U \subseteq \R$ is a finite subset of $\R$.

By construction it is seen that $\TT \subseteq \R$

By definition, $\R$ is totally ordered by $>$.

By Subset of Toset is Toset, $U$ is also totally ordered by $>$.

From Finite Totally Ordered Set is Well-Ordered, $U$ has a smallest element $m$, say.

Thus:
 * $\TT = \set {x \in \R: x > m}$

Let $n \in \R: n < m$.

Then it is seen that $n \notin \TT$ and the result follows.