Applying the Stone-Weierstrass Theorem to approximate even functions 
Let $f:[-1,1] \rightarrow \mathbb{R}$ be any even continuous function on $[-1,1]$ (i.e. $f(-x)=f(x)$ $\forall x \in [-1,1]$). Let $\epsilon>0$. Prove that there exists an even polynomial $p$ such that $$|f(x)-p(x)|< \epsilon$$ $$\forall x \in [-1,1]$$

Here, "even polynomial" means that $p(-x)=p(x)$, not simply that it has even degree.
I think I should use the Stone-Weierstrass theorem to show that the subalgebra of even polynomials, call it $\mathcal{A}$, over this interval is dense, from which the result follows immediately.
For this to work I require that $\mathcal{A}$ contains the constants (obviously true) and separates points...which is not true, unfortunately. Anyone have any hints? I would prefer hints only, rather than solutions.
Oh yes, and I should mention that the version of the Stone-Weierstrass theorem that I can use says that if a subalgebra of $C(\mathbb{R})$ contains the constants and separates points, then it is dense in $C(\mathbb{R})$.

A little late, but I'd like to add a short proof for possible future readers. The subspace $E$ of even polynomials on $[0,1]$ is dense in $C([0,1])$ by Stone-Weierstrass.  Approximate $f|_{[0,1]}$ by $p\in E$ to within $\epsilon$; then $x\mapsto p(|x|)$ on $[-1,1]$ is an even polynomial, and $|f(x)-p(|x|)|<\epsilon$ on $[-1,1]$.
