Application of Stone Weierstrass exponential polynomials

Question

It's a pretty standard result, but how does one prove, using Stone-Weierstrass, that the exponential polynomials are uniformly dense in $C([0, 2\pi])$ ($2\pi$ periodic continuous functions)? Show that the exponential polynomials contain the constants seems reasonable, but showing the separation of points, I'm unsure of.

Answer 1

Proving that the algebra generated by $\sin t$ and $\cos t$ separates points means proving that for any distinct values $a,b \in [0,2\pi)$, we have $\sin a \ne \sin b$ or $\cos a \ne \cos b$. This is immediate from the definition of the trigonometric functions.