frequency spectrum - Discreteness and periodicity in Fourier transform

Why discreteness in time / frequency domain dictates periodicity in the other frequency / time domian?

• For example the DTFT is perodic in frequency?



• Why it doesn't contain all the frequencies?


• Why the DTFS contain only finite number of frequencies?

NOTE:

I've read the derivation but sometimes i see it's like a definition or a claim in the case of DTFS.

• How we prove that equation of DTFS ? $$x[n] = \sum_{k=0}^{N-1} a_k e^{jl\omega_o n}$$