i am a bit confused. the set L = {n in Z: n < N} is the set of ALL integers less than N, whereas in the orginial question, L is presumed to be merely some (non-emtpy) set with the property that:
a in L--> a < N, which is NOT the same thing.
in other words, we are only talking about some subset of {n in Z: n < N}, and we don't know WHICH subset.
i would think it would be easier to assume our set L has no maximal element (that is, if a is in L, then there is b in L with a < b
note that a+1 ≤ b, necessarily), and from this assumption, show that some member in L must be larger than N,
which is a contradiction, so L must have a maximal element.