A real valued continuous function can be approximated by a polynomial function p(x) with rational coefficients

Question

I am unable to solve a particular question asked in my analysis quiz .

Question:Let f be a real valued continuous function on[-1,1] such that f(x)=-f(-x) for all x $\in$ [-1,1] . Show that for every $\epsilon$ >0 there is a polynomial p(x) with rational coefficients such that for every x$\in$[-1,1] ,

$|f(x)-p(x^{2})|<\epsilon$.

I thought of using stone wierestrauss theorem but It doesnot proves that all coefficients must be rational . f(-x)=f(x) will be equivalent to that there would be no terms with odd powers of x .

But how can I prove that even by taking all coefficients to be rational , I will still have uniform convergence .

Any thoughts?

Answer 1

Use Stone-Weirstrass to get an approximation by a polynomial with real coefficients. Then, approximate each coefficient by a rational number to get an approximation by a polynomial with rational coefficients.