Union is Increasing Sequence of Sets
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Theorem
Let $\sequence {D_n}_{n \mathop \in \N}$ be a sequence of sets.
Then:
- the sequence $\ds \sequence {\bigcup_{k \mathop = 1}^n D_k}_{n \mathop \in \N}$ is increasing.
Proof
We have:
- $\ds \bigcup_{k \mathop = 1}^{n + 1} D_k = D_{n + 1} \cup \bigcup_{k \mathop = 1}^n D_k$
From Set is Subset of Union, we have:
- $\ds \bigcup_{k \mathop = 1}^n D_k \subseteq D_{n + 1} \cup \bigcup_{k \mathop = 1}^n D_k$
so:
- $\ds \bigcup_{k \mathop = 1}^n D_k \subseteq \bigcup_{k \mathop = 1}^{n + 1} D_k$
So:
- $\ds \sequence {\bigcup_{k \mathop = 1}^n D_k}_{n \mathop \in \N}$ is increasing.
$\blacksquare$