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Let $(M,g)$ be a closed, compact Riemannian manifold. Let $u \in C^{k}(M)$. Can I always find a sequence of $C^\infty$ functions $\{u_n\}$ such that $u_n$ converges to $u$ in $C^k$ norm?

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up vote 6 down vote accepted

Yes, you can!

The proof of the density of $\mathcal C^\infty(M)$ in $\mathcal C^k(M)$ is a fairly technical result:
Locally (that is in open subsets of $\mathbb R^n)$ the result is proved by using convolutions with smooth functions and then, as expected, you use partitions of unity to get the global result.
Instead of approximating functions $u\in \mathcal C^k(M)$ by $\mathcal C^\infty$-functions you can even take maps $U\in \mathcal C^k(M,N)$ into another manifold $N$ and approximate them in the $\mathcal C^k$-topology by smooth maps in $\mathcal C^\infty(M,N)$.

You can find a detailed proof of this generalized result in Hirsch's Differential Topology, Theorem 2.6, page 49.

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