= Solution
For $n>m$, <independence> and $\mathbb EX_k=0$ give
$$
\lVert S_n-S_m\rVert_2^2
=\mathbb E\left|\sum_{k=m+1}^na_kX_k\right|^2
=\sum_{k=m+1}^na_k^2\longrightarrow0.
$$
Thus $(S_n)$ is a <Cauchy sequence> in the complete <L2 space> and converges in $L^2$ to some random variable $S$. The <Cauchy-Schwarz inequality> gives $\lVert Z\rVert_1\leq\lVert Z\rVert_2$ on a probability space, so the convergence also holds in $L^1$. Finally, convergence in $L^2$ implies <convergence in probability>, which implies <convergence in distribution>.
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