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Fourier Transform of Dirac Comb (Sampling Function) complete demonstration

Hello! So here's the problem: some days ago I needed to understand Dirac Comb in time domain and its Fourier Transform, which is another Dirac Comb in frecquency domain. I searched ALL THE INTERNET and never find a complete proof of that !!

So I decided to post myself a complete demonstration that anyone can understand. I even proof some auxilliary theorems that was needed in the main demonstration.

Demonstration is attached in pdf document. Hope that will help many other with same question!