Proof of First Difference Property for Fourier Series
You have the result already written down in your question. If $a_k$ are the Fourier coefficients of $x[n]$, then your third formula is the Fourier series of $y[n]=x[n]-x[n-1]$. So the Fourier series coefficients of $y[n]$ are

$$b_k=a_k-a_ke^{-jk\omega_0}=a_k(1-e^{-jk\omega_0})$$

meaning that

$$y[n]=\sum_{k=0}^{N-1}b_ke^{j\omega_0 nk},\qquad\omega_0=2\pi/N$$