Let $\bar{B}\subset\mathbb{R}^n$ be the closed unit ball. The question Are polynomials dense in $C^k\left(\bar{B}\right)$? asks if the set of all polynomials are dense in $$C^k(\bar{B})=\{ f\in C^k(B)\mid D^{\alpha}f\text{ is uniformly continuous on bounded subsets of $B$ for }|\alpha|\le k\}.$$
The accepted answer tells us to use the induction. I do not really understand how to do this.
Attempt Given Stone-Weierstrass ($C^0$), I try to show the density in $C^1$. Let $f\in C^1(\bar{B})$. We want to show that for any $\varepsilon>0$ there exists a polynomial $P$ over $\bar{B}$ such that $$ \|f-P\|+\sum_{j=1}^{n}\|D^j f-D^j P\|<\varepsilon, $$ where $\|g\|=\sup_{x\in \bar{B}}|g(x)|$ for $g\in C(\bar{B})$. For simplicity let $n=3$.
Following the accepted answer in the link above first note that we have $$ f(x)=f(x_0)+\int_{x_0}^x \langle \nabla f(\gamma(r)),\gamma'(r)\rangle dr, $$ for any nice path $\gamma$.
Now, from the density in $C^0(\bar{B})$ for any $\epsilon>0$ we can take $n$-variate polynomials $p_1,p_2,p_3$ such that $$ \|f_x-p_1\|<\epsilon,\ \|f_y-p_2\|<\epsilon,\ \|f_z-p_3\|<\epsilon, $$ where I used the notation $(f_x,f_y,f_z)$ for components of $\nabla f$.
It is tempting to define $P(x)=P_\gamma(x)=f(x_0)+\int_{x_0}^x \big(p_1(\gamma_1(r))\gamma_1'(r) + p_2(\gamma_2(r))\gamma_2'(r)+ p_3(\gamma_3(r))\gamma_3'(r)\big)dr$, as then we can let $$ \|f-P\|+\sum_{j=1}^{3}\|D^j f- p_j\|=\text{small}. $$ But the problem is we cannot say $\nabla P=(D^1P,D^2P,D^3P) =(p_1,p_2,p_3)$, can we?
My understanding is that we can say this IF $P$ is the potential of $(p_1,p_2,p_3)$. This is the case if $P_\gamma$ defined above is independent of $\gamma$, but I don't know how to say this. (Can we?)
Or do we use specific path or the domain being the ball? Or can we use more abstract discussions?
Hint: Try convolving with polynomial mollifiers like $c_n(1-|x|^2/2)^n, n=1,2,\dots,$ where $c_1, c_2, \dots $ are appropriate normalizing constants.