Resolution of Discrete Fourier Transform is 1/T - Mathematical proof?
$\frac{Fs}{N}$ 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.