Subsets in Increasing Union
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Theorem
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 increasing union of $S_0, S_1, S_2, \ldots, S_i, \ldots$:
- $\ds S = \bigcup_{i \mathop \in \N} S_i$
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 Subset Relation is 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 the definition of subset.
$\blacksquare$
Sources
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text A$: Sets and Functions: $\text{A}.2$: Boolean Operations