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## Homework Statement

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.

## Homework Equations

## 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!