Prove that if $M$ is a $C^\infty$ manifold, the space $C^r(M,\Bbb{R}^n)$ of smooth $C^r$ functions from $M$ to $\Bbb{R}^n$ is dense in $C^0(M,\Bbb{R}^n)$.
I tried looking up books of Differential Geometry for this proof, but couldn't find it. Could someone point me to a proof of this? Thanks
The idea is probably something like: use smooth and locally finite partitions of unity to reduce this to the analysis fact (on Euclidean space), which you can find in any book that treats functional analysis. (I vaguely recall that Lee's smooth manifolds books has details on the partitions of unity.)
The analysis fact is proved like this: first show simple functions are dense in the space of continuous functions. Then, show you can approximate characteristic functions of an interval by smooth functions. There are lots of epsilons.