Union is Increasing Sequence of Sets

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:


 * $\ds \bigcup_{k \mathop = 1}^{n + 1} D_k = D_n \cup \bigcup_{k \mathop = 1}^n D_k$

From Set is Subset of Union, we have:


 * $\ds \bigcup_{k \mathop = 1}^n D_k \subseteq D_n \cup \bigcup_{k \mathop = 1}^n D_k$

so:


 * $\ds \ds \bigcup_{k \mathop = 1}^n D_k \subseteq \bigcup_{k \mathop = 1}^{n + 1} D_k$

So:


 * $\ds \sequence {\bigcup_{k \mathop = 1}^n D_k}_{n \mathop \in \N}$ is increasing.