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

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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 $\R \subseteq \SS$

Aiming for a contradiction, suppose $x \in \R$ 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.

$\blacksquare$


Sources