[proofplan] An atlas on $M$ has open chart domains whose union is all of $M$. Thus the family $\{U_i\}_{i\in I}$ is an [open cover](/page/Open%20Cover) of the underlying [topological space](/page/Topological%20Space) of $M$. Compactness of $M$ says that every open cover of $M$ has a [finite subcover](/page/Finite%20Subcover), so finitely many of the chart domains still cover $M$. [/proofplan] [step:Recognize the atlas domains as an open cover of $M$] For each $i\in I$, the chart $(U_i,\varphi_i)$ has domain $U_i\subset M$, and the coordinate map is the homeomorphism $\varphi_i:U_i\to V_i$ onto the [open set](/page/Open%20Set) $V_i\subset\mathbb R^n$ named in the theorem statement. By the definition of a chart on a [topological manifold](/page/Topological%20Manifold), $U_i$ is open in $M$. Since $\{(U_i,\varphi_i)\}_{i\in I}$ is an atlas for $M$, its domains cover $M$: \begin{align*} M=\bigcup_{i\in I} U_i. \end{align*} Therefore $\{U_i\}_{i\in I}$ is an open cover of $M$. [guided] We first unpack exactly what the word “atlas” contributes. For every index $i\in I$, the pair $(U_i,\varphi_i)$ is a chart on $M$. This means that $U_i$ is an open subset of the topological space $M$, and $\varphi_i:U_i\to V_i$ is a homeomorphism onto the open subset $V_i\subset\mathbb R^n$ named in the theorem statement. The coordinate maps themselves will not be used in this argument; only their domains matter. Because the family $\{(U_i,\varphi_i)\}_{i\in I}$ is an atlas for $M$, its chart domains cover the whole manifold: \begin{align*} M=\bigcup_{i\in I} U_i. \end{align*} Thus the family $\{U_i\}_{i\in I}$ satisfies the two requirements for being an open cover of $M$: every $U_i$ is open in $M$, and their union is all of $M$. [/guided] [/step] [step:Apply compactness to extract finitely many chart domains] Since $M$ is compact and $\{U_i\}_{i\in I}$ is an open cover of $M$, the definition of compactness gives a finite subset $J\subset I$ such that \begin{align*} M=\bigcup_{j\in J} U_j. \end{align*} For each $j\in J$, the pair $(U_j,\varphi_j)$ belongs to the original atlas. Hence $\{(U_j,\varphi_j)\}_{j\in J}$ is a finite subcollection of the given atlas whose domains cover $M$, which is the required finite subatlas. [/step]