Frequency resolution of DFT
The result of an $N$-point DFT is an $N$ element complex vector.  For strictly real data, only the first $1+N/2$ bins matter, as the others are a redundant complex conjugate image.  For properly bandlimited data sampled at $F_s$, the frequency range is $0$ to $F_s/2$.  Divide $F_s/2$ by $N/2$, and you get $F_s/N$ as the frequency spacing of $1+N/2$ equally spaced DFT results, including both end points, DC and $F_s/2$.

So $F_s/N$ is the DFT result bin spacing of an $N$-point DFT.

However, the actual resolution can range from a fraction of the DFT bin spacing to $2$ or more DFT bins of separation, depending on the signal-to-noise ratio and what kind of resolution you want: frequency peak estimation, or peak separation (with a gap between peaks, typically of $3\textrm{ dB}$ or more).

In low noise (high signal-to-noise ratio), an isolated frequency peak can be resolved, by $\mathrm{Sinc}$ interpolation, to a much higher resolution than the DFT bin spacing, due to added information from the a-priori assumption (or a good guesstimate) of that signal-to-noise ratio and peak isolation.