I would like to ask if my proof of the fact that trigonometric polynomials are dense in $\mathcal C^0_{2\pi}$ is correct. Denoting by $\mathcal P_{2\pi}$ the set of trigonometric polynomials, I have to prove that $$\forall_{f\in\mathcal C^0_{2\pi}}\forall_{\varepsilon>0}\exists_{p\in\mathcal P_{2\pi}} ||f-p||_{\mathcal C^0_{2\pi}}<\varepsilon.$$
I already know two things I can use to prove this fact. Denoting by $(F_n)_n$ the Fejér kernel, I know that $$f*F_n(x)=\sum_{|k|\le n}\hat f(k)\left(1-\frac{|k|}{n+1}\right)e^{ikx},$$ so the convolution $f*F_n$ is a trigonometric polynomial for all $n$. I also know that, for each summability kernel $(u_n)_{n\in\mathbb N}$ one has $$\lim_{n\to+\infty}u_n*f=f\qquad\text{in $\mathcal C^0_{2\pi}$}.$$ So, since $(F_n)_n$ is a summability kernel, for each $\epsilon>0$ there exists $n\in\mathbb N$ such that $$||F_n*f-f||_{\mathcal C^0_{2\pi}}<\epsilon.$$ And I can choose $F_n*f$ as the trigonometric polynomial $p$ I was looking for. Is this proof OK? Thank you very much!