Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 2 / 25J / c

Convergence to a finite real limit is equivalent to the Cauchy property. The convergence set is thereforeEach set in braces is measurable, and sigma-algebras are closed under all the displayed countable operations. Completeness of $\mathbb{R}$ ensures that the Cauchy property gives a finite limit, proving the claim.