Let $\{U_1,U_2,\ldots\}$ be a cover of $\mathbb{R}^n$ by open sets. Prove that there is an open cover $\{V_1,V_2,\ldots\}$ such that
- $V_j \subset U_j$ for each $j$,
- each compact subset of $\mathbb{R}^n$ is disjoint from all but finitely many of the $V_j$.
My attempt:
First thing we noticed that any compact set $K\subset \mathbb{R}^n$ is contained in some closed ball $\mathbb{B}[0,n]=B_n.$ Now since, each of the $B_n$ are compact so for each $n\in \mathbb{N}$ there exists finitely many $U_J$'s which will cover the ball $B_n$. Now assume that $j_n$ be the smallest index such that $B_n\subset \bigcup_{j=1}^{j_n} U_j.$ Now I stuck how to construct the sets $V_j$ from these things.