[[Naïve set theory MOC]]
# The intersection of inhabited nested sets may be empty

Consider a strictly decreasing sequence $(S_n)_{n=1}^\infty$ of inhabited subsets (i.e. such that $\0 \neq S_{n} \supsetneq S_{n+1}$ for $n \in \mathbb{N}$).
Then it is possible that #m/thm/set/naïve
$$
\begin{align*}
\bigcap_{n=1}^\infty S_{n} = \0
\end{align*}
$$

> [!check]- Proof
> For example, let
> $$
> \begin{align*}
> S_{n} = \{ m \in \mathbb{N} : m \geq n \} \sube \mathbb{N}
> \end{align*}
> $$
> for $n \in \mathbb{N}$.
> Then clearly $(S_n)_{n=1}^\infty$ is strictly decreasing,
> but every $m \in \mathbb{N}$ has $m \notin S_{m+1}$.
> Therefore $\bigcap_{n=1}^\infty S_{n} = \0$.
> <span class="QED"/>

However, [[The intersection of nested inhabited Hausdorff-compact sets is inhabited]].

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