Let $\{t_0,t_1,...,t_n\}\subset [a,b]$ and $\mathscr{F}=\{q\in\mathscr{P}:q'(t_0)=q'(t_1)=...=q'(t_n)=0\}$, where $\mathscr{P}$ is the set of polynomials. Prove that $\mathscr{F}$ is $\|.\|_\infty$-dense in $C[a,b]$.
My try:
I have to prove that $\forall p\in C[a,b]$, $p\in \mathscr{F}$ or $p\in \text{lim}_{n\rightarrow \infty}p_n\in\mathscr{F}, \forall n\in \mathbb{N}$.
Let $p\in C[a,b]$. Stone-Weierstrass theorem tell us that $\exists$ a polynomial $q^*$ such that $\forall t\in\{t_0, t_1, ...,t_n\}$, we have $|p(t)-q^*(t)|<\epsilon$. Then $p\in\mathscr{F}$.
Any suggestions to prove $p\in \text{lim}_{n\rightarrow \infty}p_n\in\mathscr{F} \forall, n\in \mathbb{N}$ would be great!
According to the general Stone-Weierstraß theorem you only have to show that $\mathscr{F}$ contains the constant functions (trivial), is an subalgebra (easy to check) and that it separates points, i.e. for every $x \neq y \in [a,b]$ there is some $p\in \mathscr{F}$ such that $p(x) \neq p(y)$. I think some interpolation theorem could help there.