Show that for sequence of independent random variables $X_1,X_2,...$ $\sum_{i=1}^\infty X_i$ either converges or diverges almost surely

Question

I want to prove that for sequence of random variables $X_1,X_2,...$, that are independent either the sum $\sum_{i=1}^\infty X_i$ converges almost surely or diverges almost surely.

Any tips on how to prove this result? Particularly I have a problem as the definition for almost sure divergence is not provided.

Answer 1

Note that if $\sum_{n=0}^\infty X_n(w)$ is convergent, then $\sum_{n=k}^\infty X_n(w)$ is convergent, and vice versa.

Now, apply Kolmogorov 0-1 law.