This is a following-up question of
Proof — Weierstrass Approximation Theory for derivatives
and $f \in C^{\infty}$
Basically I wanted to explore whether there exists a sequence of polynomials $p_n$ such that $p^{(k)}_n$ converges uniformly on $[0, 1]$ to $f^{(k)}$ (the k-th derivative) for $ k = 0, 1, 2, . . . $
My idea is as follows:
Given any $k$, there exists a sequence of polynomial $q$ such that as $n\gt N$ we have $\|f^{(k)} - q_n\| \lt \epsilon.$ This is a direct consequence of Weierstrass Theorem.
We define $g(x) = f^{(k)}(x)$, and let $p_n(x) = \underbrace{\int_{0}^{x}... \int_{0}^{x}}_\text{$k$ times}q_n(s)ds + f(0)$ and we are done.
Given that such $k$ is an arbitrary positive integer, the result holds for all $k$.
Is this proof valid?