Example 1.44 In Rudin's Functional Analysis ("Exhaustion by Compact Sets")

Question

In example 1.44, Rudin introduces the space $C(\Omega)$, where $\Omega \subseteq \Bbb{R}^k$ is open. He claims that $\Omega$ can be written as a countable union of compact sets $\{K_n\}_{n \in \Bbb{N}}$ such that $K_n \subseteq \text{int } K_{n+1}$ for every $n \in \Bbb{N}$. How does one prove this?

Answer 1

Take $K_n = \left\{x \; : \; d(x, \Omega^{c} ) \geq {1 \over n}, \|x\| \leq n \right\}$.