Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 3F / a / Solution

Convergence of $\sum a_n$ implies $a_n\to 0$. Hence $0<a_n\le 1$ for all sufficiently large $n$, and then $a_n^2\le a_n$. The comparison test, with the finitely many initial terms treated separately, proves thatThis is the squares of a summable positive sequence lemma.