Hi

how do I proof the following lemma:

$\displaystyle $X_1,X_2,X_3,\cdots$$ independent r.v. with $\displaystyle $\mathbb{P}(X_n=1)=p_n$$ and $\displaystyle $\mathbb{P}(X_n=0)=1-p_n$$.

1) $\displaystyle $X_n \overset{P}{\rightarrow} X \Leftrightarrow p_n \rightarrow 0$$

2) $\displaystyle $X_n \rightarrow X a.s. \Leftrightarrow \sum_{n=1}^\infty p_n <\infty$$

thank you!