Really just the triangle inequality for p-norms: For any p>=1, a₁,...,a_n≥0 and b₁,...,b_n≥0,

((a₁+b₁)^p + ... + (a_n+b_n)^p)^1/p ≤ (a₁^p+...+a_n^p)^1/p + (b₁^p+...+b_n^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.