\begin{definition}[Definition:Increasing Union]
Let $S_0, S_1, S_2, \ldots, S_i, \ldots$ be a nested sequence of sets, that is:
:$S_0 \subseteq S_1 \subseteq S_2 \subseteq \ldots \subseteq S_i \subseteq \ldots$
Let $S$ be the set:
:$\ds S = \bigcup_{i \mathop \in \N} S_i$
where $\bigcup$ denotes set union.
Then $S$ is called the '''increasing union''' of $S_0, S_1, S_2, \ldots, S_i, \ldots$
\end{definition}