 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-1/23i/b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

u, v \in f^{- 1} (B_{ε})

, additivity gives

f (- v) = - f (v)

and

∣ f (u - v) ∣ = ∣ f (u) - f (v) ∣ \leq ∣ f (u) ∣ + ∣ f (v) ∣ < 2 ε .

Therefore

f^{- 1} (B_{ε}) - f^{- 1} (B_{ε}) \subseteq f^{- 1} (B_{2 ε}) .

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