Undergrad Prove this inequality involving metrics

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To prove the inequality |ρ(x, z) - ρ(y, u)| ≤ ρ(x, y) + ρ(z, u), one can utilize the triangle inequality effectively. The discussion suggests considering distances involving the points x, y, z, and u, particularly focusing on the relationships between these points. A sketch may help visualize the distances and their connections. Iterative applications of the triangle inequality will be crucial in establishing the proof. The approach emphasizes understanding the geometric interpretation of the metric space involved.
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For any metric ##(X, \rho)## and points therein, prove that ##|\rho(x, z) - \rho(y, u)| \leq \rho(x, y) + \rho(z, u)##.

I know that this will involve iterated applications of the triangle inequality...but I still need another hint on how to proceed.
 
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Did you draw a sketch? Consider distances to z-x+y, for example.
 

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