1. The problem statement, all variables and given/known data Let (X,ρ) and (Y,σ) be metric spaces. Define a metric d on X x Y by d((x1,y1),(x2,y2))=max(ρ(x1,x2),σ(y1,y2)). Verify that d is a metric. 2. Relevant equations 3. The attempt at a solution I proved positive definiteness and symmetry, but I am not sure how to prove the "triangle inequality" property of a metric. How many cases do we need in total, and how can we prove it? Any help is appreciated!