Really just the triangle inequality for p-norms:
For any p>=1, a1,...,an≥0 and b1,...,bn≥0,
((a1+b1)p + ... + (an+bn)p)1/p ≤
A version with integrals (instead of sums) exists too, of course.
A proof of Minkowski's inequality follows from some trickery with Hölder's inequality.