Proof that DFT does not require more than N points
Note that f() can only be sampled down to N points per period if f() is bandlimited to include only spectrum below t_period/(2*N) in frequency.  And a signal exactly periodic in N, with no frequency content at or above t_period/(2*N) in frequency can be exactly represented with N complex exponential basis vectors (Fourier).  And the inverse.

The number of degrees of freedom in N complex basis vectors or N complex DFT results matches the number of degrees of freedom in N complex time samples.