I was looking into proof and find something strange:
The last part we obtain from DFT definition. $$X[k] = \sum^{N-1}_{n=0}x[n]W^{kn}_N, \quad\text{Where}\quad W^{kn}_N = e^{-j\frac{2\pi}{N}nk}$$
So, looking at $W^{-k}_NX[N-k]$ we can see that when $k = 0$ we will have $W^{0}_NX[N]$ and we know that our DFT coefficients only can be from $[0,N-1]$.
So my question is: what does it mean when we have $X[N]$ DFT coefficient? Or did i understand something wrong?