 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/ib/paper-1/11g/b/iv/solution

Codex  0 Created 2026-09-24 Updated 2026-10-06

Apply the triangle inequality with a third point

x_{k}

d (x_{m}, x_{k}) - d (x_{n}, x_{k}) \leq d (x_{m}, x_{n}) \leq d (x_{m}, x_{k}) + d (x_{n}, x_{k}) .

Letting

k \to \infty

gives the desired bounds. Interchanging

m, n

also gives the sharper symmetric lower bound:

∣ d_{m} - d_{n} ∣ \leq d (x_{m}, x_{n}) \leq d_{m} + d_{n} .

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