Why is $e^{i2\pi Nf_snT}=1$?
Because $Tf{_s}$=1 and $N$ and $n$ are integers.  So the exponent becomes 2$\pi$ times an integer, say $K$. But $e{^{i\theta}} = \cos(\theta)+i\sin(\theta)$. Therefore $e{^{i2K\pi}} = 1$.