Let $X_n = \sum_j^n I_j$ be a sum of possibly dependent, but identically distibuted indicator random variables. If $\text{lim}_{n \rightarrow \infty} \mathbb{E}[X_n] \rightarrow \infty$, then $\text{lim}_{n \rightarrow \infty}\mathbb{P}(X_n > 0) \rightarrow 1$.
I am looking for if this is true, and if so, a hopefully simple proof.
$\mathbb{P}(X_n > 0) = 1- \mathbb{P}(X_n = 0) \geq 1-\sum_{j=1}^n \mathbb{P}(\{I_j=0\}) = 1-n\mathbb{P}(\{I_1=0\}) = 1-n(1-\mathbb{P}(\{I_j=1\})$, but I can't continue from here, even if the proof is true.