Weierstrass approximation by trigonometric polynomials

A trigonometric polynomial is a function $p$ on $\mathbb {R}$ of the form

for some $c_n \in \mathbb {C}$, $N \geq 0$. Suppose $f$ is continuous and $2\pi$-periodic on $\mathbb {R}$, and let

Show that for every $\varepsilon > 0$, there exists a trigonometric polynomial

such that

and

Prove that the trigonometric polynomials form a dense set in $C^k(\mathrm{T}^n)$ for any $k$.

Prove that the space of trigonometric polynomials is dense in $\mathcal{E}(\mathrm{T})$.

Every continuous function that is periodic with period $2\pi$ can be uniformly approximated by trigonometric polynomials (see §2 of Chapter 1, Part VI) to any assigned degree of accuracy. (Weierstrass' approximation theorem.)

Let $f$ be a continuous real-valued function on $\mathbb {R}$ that is periodic with period $2 \pi$. For $\epsilon > 0$, show that there are real numbers, $c_{0}, a_{1}, \ldots , a_{n}, b_{1}, \ldots , b_{n}$, such that