Subsets in Increasing Union

Theorem
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 increasing union of $$S_0, S_1, S_2, \ldots, S_i, \ldots$$:
 * $$S = \bigcup_{i \in \N} S_i$$

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

Proof
Let $$k \in \N$$.

Let $$j \ge k$$.

Then by as many applications as necessary of Subsets Transitive, we have:
 * $$S_k \subseteq S_j$$

Now $$s \in S$$ means, by definition of set union, that:
 * $$\exists S_k \subseteq S: s \in S_k$$

Then from above:
 * $$j \ge k \implies S_k \subseteq S_j$$

it follows directly that:
 * $$\forall s \in S: \exists k \in \N: \forall j \ge k: s \in S_j$$

from definition of subset.