I'm taking an introductory course in Fourier analysis and I'm trying to solve the following problem
Prove that the Fourier series of a continuously differentiable function $f$ on the circle is absolutely convergent.
[Hint: Use the Cauchy-Schwarz inequality and Parseval's identity for $f'$.]
I solved it like this $$\sum_{n=-\infty}^{\infty} |\hat{f}(n)| \le \sum_{n=-\infty}^{\infty} |\hat{f}(n)|^2 = \dfrac{1}{2\pi}\int_{0}^{2\pi}|f(x)|^2 < \infty$$
I'm thinking that the equality holds because of Parseval's identity and that the integral is bounded because $f$ is bounded because it's continuously differentiable on the circle.
The problem is that I'm pretty sure that this solution is wrong since I haven't used the information in the hint or the fact that $f$ is differentiable. Indeed, if this solution is correct then this should hold for any continuous $f$? I'm just not sure why this solution is wrong.
The correct solution is available here (page 30)