Question in more precise format:
If $f$ is a continuous function on $\Bbb R$ with period 2$\pi$ and if $\epsilon>0$, then there is a trigonometric polynomial $P$ such that $|P(x)-f(x)|<\epsilon$ for all real $x$.
Given: A trigonometric polynomial is a finite sum of the form $$ P(x) = a_o + \sum_{n=1}^N (a_n \cos nx + b_n \sin nx) $$ where $x$ is real and $a_0,a_1...a_N,b_1,...b_N$ are complex numbers
Hint given: Identifying $x$ and $x+2\pi$ regard 2$\pi$-periodic functions on $\Bbb R$ as functions on the unit circle $\Bbb T$ by means of the mapping $x\to e^{ix}$.
I am just starting with the subject and would thus like to see a rigorous proof of this so as to get acquainted with the proof methodology.