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I heard of a theorem saying that each $\mathcal{C}^k$-manifold with $k\geq 1$ can be made into a smooth manifold, i.e. $\mathcal{C}^{\infty}$ (by restriction of the atlas).

However, I cannot find this theorem anywhere. Can anyone point me in the write direction (book, paper, webpage, ...) and/or give me the proof?

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This result can be found in Hirsch's Differential Topology. More precisely, Theorem $2.9$ of section $2$, chapter $2$ which I have reproduced below.

Theorem: Let $\alpha$ be a $C^r$ differential structure on a manifold $M$, $r \geq 1$. For every $s$, $r < s \leq \infty$, there exists a compatible $C^s$ differential structure $\beta \subset \alpha$, and $\beta$ is unique up to $C^s$ diffeomorphism.

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