How to prove that the subset of polynomials $A=\{\sum_{k=0}^n{a_kx^{2k}}:a_i\in\mathbb{R},n\in\mathbb{N} \}$ is dense in $C[0,1]$

Question

I have $$A=\left\{\sum_{k=0}^n{a_kx^{2k}}:a_i\in\mathbb{R},n\in\mathbb{N} \right\}$$ and I have to prove that A is dense in $C[0,1]$ with respect to the supremum norm.

My efforts in trying to solve the problem:
I showed that the set of polynomials is dense in $C[0,1]$ by using Bernstein's polynomials, so if I can prove $A$ is dense in the set of polynomials I can conclude that A is dense in $C[0,1]$.

But trying to do so, I am in a dead end...any help?

Answer 1

Let $f\in C([0,1])$ and consider $g(x)=f(\sqrt{x})\in C([0,1])$. By Weierstrass approximation theorem, there is a sequence of polynomials such that $\|p_n-g\|_{\infty}\to 0$. This gives $\|p_n(x^2)-g(x^2)\|_{\infty}\to 0$, but $p_n(x^2)$ belongs to $A$ and $g(x^2)=f(x)$, so $A$ is dense in $C([0,1])$.