Let Q(n) denote "Forall k in N(k<n implies P(k))". Then Q(n) is a property on N
If Forall n in N(Q(n) implies P(n)):
    Forall n in N(Q(n) implies P(n))
    There is no k in N such that k<0, hence k<0 and P(0) are both empty
    Thus Q(0) by vacuous truth
    Given n in N:
        If Q(n):
            Q(n)
            Given k in N:
                If k<n+1:
                    If k>n:
                        n<k<n+1
                        No such k
                        FALSE
                    k<=n