Tail of Decreasing Sequence of Sets is Decreasing
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Theorem
Let $X$ be a set.
Let $\sequence {E_n}_{n \mathop \in \N}$ be a decreasing sequence of subsets of $X$.
Then for each $m \in \N$ we have:
- $\sequence {E_{n + m} }_{n \mathop \in \N}$ is a decreasing sequence of sets.
Proof
Since $\sequence {E_n}_{n \mathop \in \N}$ is an decreasing sequence of sets, we have:
- $E_{n + 1} \subseteq E_n$ for each $n \in \N$.
Swapping $n$ for $n + m$, this in particular gives:
- $E_{n + m + 1} \subseteq E_{n + m}$ for each $n \in \N$.
So $\sequence {E_{n + m} }_{n \mathop \in \N}$ is a decreasing sequence of sets.
$\blacksquare$