Consider an assertion made on Wolfram about the Fourier series:
Question: How can one just assert that $f(x)$ is a real-valued function and not a complex-valued one? That is, how can one just assert that
$$ f(x) = \sum_{n=-\infty}^\infty A_n e^{inx}? $$
If $A_n \in \mathbb{C}$ and $e^{inx} \in \mathbb{C}$, then wouldn't it be an extraordinary coincidence if $f(x)$ happened to be in $\mathbb{R}$ as well?