Having real trouble solving this problem, I know there's probably something obvious that I'm missing but it's driving me mad!
Exercise 1.3. Let $f \in C(\mathbb{R} / \mathbb{Z})$. For $y > 0$ let $$\omega(y) = \int_0^1 |f(t + y) - f(t)| dt.$$ Show that the Fourier coefficients $c_k$ of $f$ satisfy $$|c_k| \le \frac 1 2 \omega\left(\frac 1 {2k}\right)$$ for $k \ne 0$.