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$$

f(\theta) = \sum_{n = -\infty}^{\infty}A_ne^{in\theta}

$$

Since $f$ is bounded, $|f| < M = |z|\in\mathbb{C}$. Since it could be $\mathbb{C}$, $M$ would be the modulus correct?

We know that the modulus of $e^{in\theta}$ is 1 so $|f| = \sum\limits_{n = \infty}^{\infty}|A_n|$.

How to finish it?