Definition:Increasing Union

Definition
Let $$S_0, S_1, S_2, \ldots, S_i, \ldots$$ be sets such that:
 * $$S_0 \subseteq S_1 \subseteq S_2 \subseteq \ldots \subseteq S_i \subseteq \ldots$$

that is, each set is contained in the next as a subset.

Let $$S$$ be the set:
 * $$S = \bigcup_{i \in \N} S_i$$

where $$\bigcup$$ denotes set union.

Then $$S$$ is called the increasing union of $$S_0, S_1, S_2, \ldots, S_i, \ldots$$

From Subsets in Increasing Union, we have that:
 * $$\forall s \in S: \exists k \in \N: \forall j \ge k: x \in S_j$$