Problem 2

Exercise 3.2 (Squeeze theorem for Euclidean sets)   Let $K\subset U \subseteq\mathbb{R}^n$ where $K$ is compact. Find $V$ such that $K\subseteq V\subseteq \overline{V} \subseteq U$ and $\overline{V}$ is compact.

Proof. Cover $K$ with balls interior to $U$ and extract a finite subcover.

$$K \subseteq \bigcup_{x\in K} B_{\epsilon_x}(x)\cap K$$ (3.14)

$$\subseteq \bigcup_{i=1}^n B_{\epsilon_i}(x_i) \cap K$$ (3.15)

Define a family of open sets to help us find an appropriate set $V$. Set

$$V_\eta = \bigcup_{i=1}^n B_{\epsilon_i-\eta}(x_i) \subseteq \bigcup_{i=1}^n B_{\epsilon_i}(x_i) \subseteq U$$ (3.16)

Now refine. For each $x\in K$, select $\epsilon>0$ and $x_i$ such that $\vert x-x_i\vert<\epsilon<\epsilon_i$ by density. Then setting $\eta_x < \epsilon_i-\epsilon$, we can realize another open cover

$$K \subseteq \bigcup_{x\in K}V_{\eta_x}$$ (3.17)

$$\subseteq \bigcup_{j=1}^m V_{\eta_j} = V$$ (3.18)

Setting $V$ as indicated, we can tell $\overline{V}\subseteq U$, as desired. $\qedsymbol$