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:

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

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.

$\blacksquare$