By the Pauli exclusion principle, no two fermions can be in the same state. Additionally, at zero temperature the enthalpy of the electrons must be minimal, meaning that they cannot change state. If, for a particle in some state, there existed an unoccupied lower state that it could occupy, then the energy difference between those states would give the electron an additional enthalpy. Hence, the enthalpy of the electron would not be minimal. Therefore, at zero temperature all the lowest energy states must be saturated. For a large ensemble the Fermi level will be approximately equal to the chemical potential of the system, and hence every state below this energy must be occupied. Thus, particles fill up all energy levels below the Fermi level at absolute zero, which is equivalent to saying that is the energy level below which there are exactly N states.