TITLE: Theorem by Whitney
QUESTION [5 upvotes]: For $0<k<\infty$ and any $n$-dimensional $C^k$ manifold the maximal atlas contains a $C^\infty$ atlas on the same underlying set by a theorem due to Whitney. Could someone please point me to where I can find the theorem and a proof thereof?

REPLY [3 votes]: Morris Hirsch's Differential Topology text contains a proof.  I don't have the text here, but it appears fairly early in the textbook.  The key idea is in his theorem on increasing the order of differentiability of a function.