Orthogonal Random Variable $\mathcal{L}^{2}$ (Hard?)

Question

I am self-studying Probability Theory and for me the following problem is challenging.

Answer 1

The idea is that $S=\sum_{j=1}^\infty X_j$. To prove that the series converges, you need to show that the tails go to zero in the $L^2$-norm. So $$ E[(\sum_{j=m}^r X_j)^2]=E[\sum_{j,k=m}^r X_jX_k]=\sum_{k,j=m}^rE(X_jX_k)=\sum_{j=m}^rE(X_j^2). $$ The hypothesis gives you that the latter sums can be made arbitrarily small.