Nested_Compact_Set_Property

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Definition

If $K_1 \supseteq K_2 \supseteq K_3 \supseteq \cdots$ is a nested sequence of nonempty compact sets, then the intersection $\bigcap_{n=1}^{\infty} K_n$ is nonempty.

Proof

For each $n$, choose $x_n \in K_n$. Since $x_n \in K_1$ for all $n$ and $K_1$ is compact, the sequence $(x_n)$ has a convergent subsequence $x_{n_k} \to x$ with $x \in K_1$.

For any fixed $m$, the subsequence $(x_{n_k})$ is eventually in $K_m$ (since $n_k \geq m$ implies $x_{n_k} \in K_{n_k} \subseteq K_m$).

Since $K_m$ is closed, $x \in K_m$.

Thus $x \in K_m$ for all $m$, so $x \in \bigcap_{n=1}^{\infty} K_n$.