The Problem states:
If $c_n$ are the Fourier coefficients for $f$ using the orthonormal set $\{f_n\}^{\infty}_{n=1}$, show:
$$\Bigg \langle \sum^{\infty}_{n=1} c_n f_n , f - \sum^{\infty}_{n=1} c_n f_n \Bigg \rangle = 0.$$
I am assuming $f$ equals $\sum^{\infty}_{n=1} f_n$ not, $\; \sum^{\infty}_{n=1} c_n f_n$ since the right side of the inner product would immediately reduce to zero? My guess is that it isn't that simple.
From what I understand, the $f_n$'s are elements of a set of orthonormal "vectors" which form a basis. To write down $f$ in this basis, each basis "vector" must have the appropriate coefficient $c_n$. On the other hand, I do not know if we are to assume that the sum $\sum_n c_n f_n$ does converge to $f$ or not.
I just realized they have a general definition for Fourier Coefficients in Theorem 3.6 in the textbook so I'll give your suggestion a try.