Resolution of Discrete Fourier Transform is 1/T - Mathematical proof? In many articles I see that the frequency resolution of the Discrete Fourier Transform (DFT) equals Fs/N where Fs is the sampling rate and N is the total number of samples. Fs/N is equivalent to 1/T where T is the total time the signal was sampled. 

So the longer the time of sampling took place the better the resolution of the spectrum (allowing to separate different tones)

I'm looking for the mathematical proof of the 1/T resolution as I do not see it directly from the definition of the DFT.
$\frac{Fs}{N}$ is not equivalent to $\frac{1}{T}$, since $Fs = \frac{1}{T}$,  $\frac{1}{T N} \neq \frac{1}{T}$ for $N&gt;1$ (as it is implied here of course).

The point about the resolution of the Discrete Fourier Transform follows from its definition where an $N$ point transform, "divides the circle" in $N$ discrete points. As you can see $k n$ (the discrete frequency and sample "instance" variables) are multiplied by the constant $\frac{i 2 \pi}{N}$.

Here, it is still possible to "sample" a non-integer harmonic (say $k = 1.267$) but that would not improve the situation if you only have a limited number of true $n$ (samples).

Hope this helps.