DFT shift theorem proof
For an $N$-periodic sequence $y[n]$, the sum over one period is always the same, no matter how you choose the first index:

$$\sum_{n=0}^{N-1}y[n]=\sum_{n=\Delta}^{N-1+\Delta}y[n]$$

for any integer $\Delta$. Try to verify that the sequence you sum over is indeed $N$-periodic.