We want to estimate the max norm of the square matrix $M:=\frac1{\sqrt{N}}(m_{r,s})_{r,s\in[0,N-1]} \in\mathbb{C}^{N\times N}$ given by $m_{r,s}=e^{irs\frac{2\pi}N} \,\forall r,s\in [0,N-1]$
Prove that $|m_{r,s}|\le 1$ for all $r,s\in [0,N-1]$.
As in the previous part of the exercise I already proved $M$ is a unitary matrix (using geometric series), I'm quite stuck with this part. Obviously taking $r=0$ or $s=0$ we immediately get $|m_{r,s}|=1$, but what about the other values? Differently I found that $e^{2\pi i}=1$ and so $m_{r,s}=e^{irs\frac{2\pi}N}= \left({e^{2\pi i}} \right)^{\frac{rs}N}=1^{\frac{rs}N}=1 \,\, \,\forall r,s\in [0,N-1]$. Is this to exclude?
Every complex number $z=a+ib$ with $a,b\in\mathbb{R}$ can also be written in the polar form $z=re^{i\theta}$ with $r\in[0,\infty)$ and $\theta\in[0,2\pi)$. The relationship between the two expressions is $a=r\cos\theta$ and $b=r\sin\theta$ which implies that $|z|=\sqrt{a^2+b^2}=r$. Therefore, any complex number of the form $e^{i\theta}$, such as $m_{r,s}$ in your exercise, has unit absolute value.