Stone-Weierstrass theorem. Let $X$ be a compact Hausdorff space, let $\mathscr{A}$ be the algebra of real valued continuous functions on $X$ and let the topology in $\mathscr{A}$ of uniform convergence. Then a necessary and sufficient condition that a subalgebra $\mathscr{F}$ of $\mathscr{A}$ be dense in $\mathscr{A}$ is that
(1) for each $x$ in $X$ there is an $f$ in $\mathscr{F}$ with $f(x)\neq 0$,
(2) for $x\neq y$ in $X$ there is an $f$ in $\mathscr{F}$ with $f(x)\neq f(y)$,
Nachbin's theorem gives an analog for Stone–Weierstrass theorem for algebras of real valued $C^r$ functions on a $C^r$ manifold.
Nachbin's theorem. Let $M$ be a manifold of class $C^r$, let $\mathscr{A}$ be the algebra of real valued functions of class $C^r$ on $M$ and let the topology in $\mathscr{A}$ be that of uniform convergence on compact sets up to the derivatives of order $r$. Then a necessary and sufficient condition that a subalgebra $\mathscr{F}$ of $\mathscr{A}$ be dense in $\mathscr{A}$ is that
(1) for each $x$ in $M$ there is an $f$ in $\mathscr{F}$ with $f(x)\neq 0$,
(2) for $x\neq y$ in M there is an $f$ in $\mathscr{F}$ with $f(x)\neq f(y)$,
(3) for $x$ in $M$ and a tangent vector $v$ in $T_x(M)$ there is an $f$ in $\mathscr{F}$ such that $Df(x)v\neq 0$.
The only references (with the respective proof) that I found for Nachbin theorem were the two below:
Llavona, José G. Approximation of continuously differentiable functions. North-Holland Mathematics Studies, 130. Notas de Matemática [Mathematical Notes], 112. North-Holland Publishing Co., Amsterdam, 1986. xiv+241 pp. ISBN: 0-444-70128-1
L.Nachbin. Sur les algèbres denses de fonctions diffèrentiables sur une variètè, C.R. Acad. Sci. Paris 228 (1949) 1549-1551
I would like more accessible references to quote them in a future article.
By "accessible" I mean articles that can be downloaded by online databases or books published in the last two decades which are possible to buy online.
Thank you in advance.