Definition:Increasing Union

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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 $S_0, S_1, S_2, \ldots, S_i, \ldots$

Also see

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