 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/ii/paper-2/25j/c/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

Convergence to a finite real limit is equivalent to the Cauchy property. The convergence set is therefore

k = 1 ⋂ \infty N = 1 ⋃ \infty m, n \geq N ⋂ {∣ f_{m} - f_{n} ∣ \leq 1/ k} .

Each set in braces is measurable, and sigma-algebras are closed under all the displayed countable operations. Completeness of

R

ensures that the Cauchy property gives a finite limit, proving the claim.

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