Relative Complement of Decreasing Sequence of Sets is Increasing
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Theorem
Let $X$ be a set.
Let $\sequence {E_n}_{n \mathop \in \N}$ be a decreasing sequence of sets in $X$.
Then $\sequence {X \setminus E_n}_{n \mathop \in \N}$ is an increasing sequence of sets in $X$.
Proof
Since $\sequence {E_n}_{n \mathop \in \N}$ is decreasing, we have:
- $E_{n + 1} \subseteq E_n$ for each $n \in \N$.
From Relative Complement inverts Subsets, we then have:
- $X \setminus E_n \subseteq X \setminus E_{n + 1}$ for each $n \in \N$.
So $\sequence {X \setminus E_n}_{n \mathop \in \N}$ is an increasing sequence of sets in $X$.
$\blacksquare$