Convergence of random variables - LLN (proof)

Question

I need help.

Prove: For arbitrary $\left\{ X_{n}\right\} $, if $\sum_{n}\mathbb{E}\left[X_{n}\right]<\infty$ then $\sum_{n}X_{n}<\infty$ converge absolutely almost everywhere.

Answer 1

What you say is not true. It's true if you assume that $X_n\ge0$. It's true if you assume that $\sum\Bbb E[|X_n|]<\infty$.

Let's assume that $\sum\Bbb E[|X_n|]<\infty$. The Monotone Convergence Theorem shows that $$\Bbb E\left[\sum|X_n|\right]<\infty.$$Hence $\sum|X_n|<\infty$ almost surely.