Let $X_n \geq 0$ be independent for $n \geq 1$. Then the following are equivalent:
(i) $\sum_{n=1}^\infty X_n$ converges a.s.
(ii)$\sum_{n=1}^\infty {[P(X_n >1) + E[ X_n I_{X_n \leq 1} ] }] < \infty $
(iii)$\sum_{n=1}^\infty E [ X_n / (1+ X_n)] < \infty $
How could we show the above? I tried using Kolmogorov's Three-Series Theorem but I was unable to show it.