= Solution
Set $M_n=\sum_{j=0}^nX_j$ and $\mathcal F_n=\sigma(X_0,\ldots,X_n)$. The <independence> and zero <expected values> give $\mathbb E[M_{n+1}\mid\mathcal F_n]=M_n$, so $(M_n)$ is a <martingale>. The <variance additivity for independent random variables> gives
$$
\mathbb E[M_n^2]=\sum_{j=0}^n\mathbb E[X_j^2]\leq V:=\sum_{j=0}^\infty\mathbb E[X_j^2]<\infty.
$$
The <Cauchy-Schwarz inequality> implies $\sup_n\mathbb E|M_n|\leq\sqrt V$. Applying the <martingale convergence theorem> from (b),
$$
\boxed{\sum_{j=0}^{\infty}X_j\ \text{converges almost surely}.}
$$
In fact the <L2 martingale convergence theorem> also gives <convergence in L2>. Independently, for $n>m$ the same <variance additivity for independent random variables> yields $\mathbb E|M_n-M_m|^2=\sum_{j=m+1}^n\mathbb E[X_j^2]\to0$, confirming that the partial sums form a <Cauchy sequence> in $L^2$.
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