Let $X_1$ , $X_2$ , $X_3 $ ... be independent random variables, with $P[X_n=1]=p_n $and $ P[X_n=0]=1-p_n $

Question

Let $X_1$ , $X_2$ , $X_3 $ ... be independent random variables, with $P[X_n=1]=p_n $and $ P[X_n=0]=1-p_n$. Show that

(a) $X_n $ converges to $0$ in Probability iff lim $p_n$ goes to $0$.

(b) $X_n $ converges to $0$ almost surely iff $\sum_{n=1}^{\infty} X_n $ converges.

Initially I found it is quit obvious by using definition.

For (a) Let assume $X_n $ converges to $0$ in Probability then by definition $P( |X_n| - 0 > \epsilon ) \to 0$ as $n \to \infty$, and that means $P(X_n=1) = p_n \to 0$ and vice versa.

For (b) Borel cantelli lemmas work right?

I really want know where I got wrong? If you provide your idea then I would be nice.