Almost Sure convergence of sum of independent random variables
Solution 1:
Here is a simple proof. By monotone convergence theorem: $$ \sum_j E|X_j| = E \big[ \sum_{j} |X_j| \big]. $$ It follows from the assumption that $E \big[ \sum_j |X_j| \big] < \infty$. Any random variable which has finite expectation should be finite almost surely. Thus, $\sum_j |X_j| < \infty$ almost surely. But absolute convergence for series implies convergence, hence $\sum_j X_j$ converges almost surely.