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Suppose we've a sequence $f_{n}(x)$ of $C^{\infty}$ functions, with compact support in $R^{n}$. Define $h_{n}(x) = \inf\{f_{n}(x), f_{n+1}(x),....\}$. Is this $h_{n}(x) \in C^{\infty}$? Thanks

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    The question makes no sense as it stands, as the functions are defined on different domains $\mathbb R^n$. Presumably this was meant to be $\mathbb R^k$ with a different variable $k$?2011-09-29

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