Let $M \subset \mathbb{R}^N$ be a smooth manifold of dimension $m$ where $m<n$. Suppose $f : M \to \mathbb{S}^n$ is a continuous map. How can I prove that for every $\varepsilon >0$ I can find a smooth function $f_{\varepsilon}:M \to \mathbb{S}^n$ with $\sup\limits_{x \in M} \|f_{\varepsilon}-f\| < \varepsilon$?
I guess that might help the theory of "partition of unity" to construct such a $f_{\varepsilon}$ but I do not know it and I would like to prove the assertion without it.
I started from the case when $M$ is compact and I easily concluded using Stone-Weiestrass result. But how can I conclude in general case?
I will appreciate any help or suggestions.
Using a smooth partition of unity, it suffices to consider the case that $M$ is $\Bbb R^n$ or even just the unit ball therein, and that the image of $M$ is contained in a hemisphere, say. Folding $f$ with a smooth kernel with tiny support (and normalizing) will then make $f$ smooth.