Inner Product of Fourier Series with Orthonormal Set

Question

I am stuck on where to go next for this problem. I would also like to know if my steps so far are correct.

If $c_n$ are Fourier coefficients of $f$ and $f_n$ is an orthonormal set, show that the inner product $$\left<\sum_{n=1}^{N}c_nf_n,f-\sum_{n=1}^{N}c_nf_n\right> = 0$$

This is what I have so far:

$$\left<\sum_{n=1}^{N}c_nf_n,f-\sum_{n=1}^{N}c_nf_n\right> = \left<\sum_{n=1}^{N}c_nf_n,f\right>+\left<\sum_{n=1}^{N}c_nf_n,-\sum_{n=1}^{N}c_nf_n\right> =$$ $$\left<\sum_{n=1}^{N}c_nf_n,f\right>-\left<\sum_{n=1}^{N}c_nf_n,\sum_{n=1}^{N}c_nf_n\right> = \left<\sum_{n=1}^{N}c_nf_n,f\right>-\sum_{n=1}^{N}\sum_{m=1}^{N}c_nc_m\left<f_n,f_m\right>=\sum_{n=1}^{N}c_n\left<f,f_n\right>-\sum_{n=1}^{N}\sum_{m=1}^{N}c_nc_m(1)=\sum_{n=1}^{N}c_nc_n-\sum_{n=1}^{N}\sum_{m=1}^{N}c_nc_m.$$

Answer 1

Pick $f_k$ with $k \in \{1,...,N\}$, then $\langle f_k , f-\sum_{n=1}^N c_n f_n \rangle = \langle f_k , f \rangle - \langle f_k , \sum_{n=1}^N c_n f_n \rangle = \langle f_k , f \rangle - \sum_{n=1}^N c_n\langle f_k , f_n \rangle = c_k -c_k = 0$.

Answer 2

The last part is not quite right. It should be $\big \langle \sum_{n=1}^N c_nf_n,f \big \rangle - \sum_{n=1}^N \sum_{m=1}^N c_nc_m \langle f_n,f_m \rangle= \sum_{n=1}^N c_n \langle f, f_n \rangle - \sum_{n=1}^N c_n^2$