Prove that the DFT Matrix is Unitary

Note that $W_{ij} = \sum_{k=0}^{N-1} \omega^{(j-i)k}$. Let $j-i\neq0$, then $\omega^{(j-i)}\neq 1$ itself is another $N$-th root of unity and let's call it $\omega_0$. Hence, what you get is $$W_{ij} = \sum_{k=0}^{N-1} \omega_0^{k} = \frac{1-\omega_0^N}{1-\omega_0} = 0.$$