Tail of Decreasing Sequence of Sets is Decreasing

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.

$\blacksquare$