Cauchy–Schwarz memberi \(\sum a_ib_i\leq(\sum a_i^2)^{1/2}(\sum b_i^2)^{1/2}\text{,}\) sehingga \(d(x,z)^2\leq(d(x,y)+d(y,z))^2\) dan karenanya \(d(x,z)\leq d(x,y)+d(y,z)\text{.}\)