Prove the following DFT for a periodic sinc sequence or ideal lowpass filter I need to prove the following DFT for a periodic  sequence or ideal lowpass filter:

$\sqrt{\frac{p}{m}}\frac{sinc(\frac{\pi np}{m})}{sinc(\frac{\pi p}{m})} \stackrel{DFT}{\longleftrightarrow} \begin{cases}
                    \sqrt{\frac{m}{p}},\: if \: |k-\frac{m}{2}|\ge \frac{p-1}{2}\\
                     0, \: otherwise
                 \end{cases}$

But actually, I do not know how to do it. Maybe someone gives me at least a hint?
Hints:

1) Definition of the unnormalized DFT:

$$ X[k] = \sum_{n=0}^{N-1} x[n] e^{-i\frac{2\pi}{N}kn}  $$

2) Geometric Summation Formula:

$$ \sum_{n=0}^{N-1} r^n = \frac{1-r^N}{1-r} $$

3) Exponential form of Sine:

$$ \sin( \theta ) = \frac{ e^{i\theta} - e^{-i\theta} }{ 2i } $$

4) Definition of unnormalized sinc:

$$ \operatorname{sinc}(x) = \frac{\sin(x)}{x} $$