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:
 * $\displaystyle S = \bigcup_{i \in \N} S_i$

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:
 * $\forall s \in S: \exists k \in \N: \forall j \ge k: x \in S_j$