Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 26J / b / Solution

The hypothesis says exactly that $X_n\rightharpoonup 0$ in the Hilbert space $L^2$. We give the constructive Weak Banach–Saks theorem in a Hilbert space argument. Choose $n_1<n_2<\cdots$ inductively so that, for $Y_k=X_{n_k}$,This is possible because each earlier $Y_j$ is an admissible test variable in the weak-convergence assumption. Since $\Vert Y_k{\Vert }_2\le 1$,Therefore