scaling - why should I scale the fft using 1/N?

I am writing a report, and my advisor asked me to explain why I scale the fft by a factor 1/N (where N is the length of the array).

I used to use the scaling convention of multipling the fft by the time increment (dt), this convention was good for me, because it ensures the check of the Parseval theorem. Unfortunately I had discussion with one of my advisors, because, since this convention does not give you the right amplitude, he thinks it is wrong. As I have read online, there are not right convention or wrong convention. If I use the factor 1/N the amplitude is checked, if I use the factor dt then the parseval identity is preserved. I have 2 question now:

1. Why, doing the fft, I cannot have both: amplitude checked and energy checked?


2. I have already demonstrated (in my report) that if I scale the fft by a factor 1/N, I obtain the right amplitude, since the first value of my fft is equal to the time average of my function. Now I would like to show with formulas why this convention gives me the right amplitude? I have already searched online and here on the forum, but I did not find any good answer that explain every passage.