[tex]Let X_1, X_2, ... be independent with P(X_n = 1) = p_n and P(X_n = 0) = 1- p_n. Show that (i) X_n \rightarrow 0 in probability iff p_n \rightarrow 0, and (ii) X_n \rightarrow 0 a.s iff \sum p_n \rightarrow \infty[\MATH]

Attemp

(i) I used Chebyshev's ineq for X_n \rightarrow 0 in probability implies p_n \rightarrow 0. Can someone give hint for the other direction?

(ii)\sum p_n \rightarrow \inft implies X_n \rightarrow 0 a.s by Borel-Cantelli Lemma. What should i do for the other direction?

Thank you in advanced!!!