# Convergence of sequence of independent random variables

Let $$x_n$$ be a sequence of independent random variables such that $$P(x_n = 0) = 1 - \frac{1}{n}$$

1) Does $$x_n$$ converge to $$0$$ almost surely

2) Does $$x_n$$ converge to $$0$$ in probability

3) Does $$x_n$$ converge to $$0$$ in $$L_p$$

It is clear for me that answer for 2) is yes: $$P(|x_n| > \epsilon) \leq P(x_n \neq 0)$$. Since the sequence of the last elements converges to $$0$$ we can conclude that $$x_n$$ converges to $$0$$ in probability.

I need some hints how to approach the remaining two questions. Thanks!

For question $$1$$, consider a sequence of independent random variables $$(X_n)$$ such that $$P(X_n=0)=1-n^{-1}$$ and $$P(X_n=1)=n^{-1}$$. Argue using Borel-cantelli that $$X_n=1$$ infinitely often with probability one and hence the sequence does not converge a.s. to $$0$$.
For question $$3$$, consider the sequence of random variables $$(X_n)$$ such that $$P(X_n=0)=1-n^{-1}$$ and $$P(X_n=n^{1/p})=1/n$$ and show that $$E|X_n|^p=1$$ for all $$n$$.