I am trying to solve this problem:
Given $X$ compact manifold, prove that any continuous function $f: X \to S^p$ can be uniformly approximated by differentiable functions.
So, what I thought was this, it all comes down to prove for functions of the form $f: \mathbb{R}^n \to \mathbb{R}$. Because, since $X$ is compact, you have a finite family $A=\{(U_1,\phi_1),...,(U_m,\phi_m)\}$ of local charts that cover $X$, lets say $X$ is an $n$-dimensional manifold, and since the $U_i$ are finite you can suppose the $\tilde{U_i}=\phi(U_i)$ disjoint.
Then the map $\tilde{f}: \cup_{1}^m{\tilde{U_i}} \to S^p \subset \mathbb{R^{p+1}}$ given by $\tilde{f}(x)=f(\phi^{-1}(x))$ is a continous mapping.
So, if I can uniformly approximate to $\tilde{f}$ by differentiable functions $\{g_i\}_{i \in \mathbb{N}}$, lets say close enough to $\tilde{f}$, such that they are nowhere zero and with a little subtlety you can assure that given $z=\phi_i(x)$ and $y=\phi_j(x)$ for $i,j \in \{1,...,m\}$, you have $g_l(z)=g_l(y) \forall l \in \mathbb{N}$. (This is done by approximating locally in each $\tilde{U_i}$)
Then taking $h_i=\frac{g_i}{\mid\mid g_i\mid\mid}$, and $H_i=h_i \circ \phi_j(x)$ where $\phi_j$ is an appropiate chart that depends on $x$, by the subtlety above this fucntions do not depend on the choice of chart, then these functions uniformly approximate to $f$.
To approximate $\tilde{f}$, all I really got to do is approximate its coordenate functions $\tilde{f}_i:\cup_{1}^m{\tilde{U_i}} \to \mathbb{R} $. Since the $\tilde{U_i}$ are disjoint it actually comes to approximate $F_j=\tilde{f}\restriction_{\tilde{U_j}}$, and just paste the approximations.
Keeping in mind that since $f(X)\subset S^p$ is a bounded function I can work in the space of bounded functions: $\mathbb{R^n}\to \mathbb{R^{p+1}}$.
All I need to do is prove the last part. Any ideas or errors you can find in this reasoning would be appreciated, thanks in advanced.
I think that using the Weierstrass Approximation Theorem should give the desired result: https://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem
However, I think you should proceed a little differently in the beginning. Namely, cover $X$ by compact sets $C_1, \dots , C_k$ such that each $C_i$ is homeomorphic to a closed rectangle in $\R^n$ and moreover each $C_i$ is contained entirely within some chart. Then apply the Weierstrass approximation theorem. (I think it is necessary to cover by compact sets because the Weierstrass approximation theorem requires the interval to be closed.) Since there are finitely many components (manifolds are finite-dimensional) and finitely many $C_i,$ you will get a uniform approximation (since you can pick the maximum of some finitely many big N's.