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
One way to solve this problem is to use Whitney embedding theorem: Embed $M$ in $R^{2n+1}$ via a proper map $i: M\to R^{2n+1}$ so that $0\notin i(M)$. Then the function $h: x\mapsto ||i(x)||$ is a smooth exhaustion function on $M$. (Properness of $i$ translates to the property that $h$ is an exha...