# Probability - almost sure convergence

1. May 11, 2012

### Gregg

1. The problem statement, all variables and given/known data

We have $\mathbb{P}(X_n = 1) = p_n$ and $P(X_n=0) = 1-p_n$ the question is about almost sure convergence. i.e. does $X_n \overset{a.s.}{\longrightarrow} 0$ if $p_n = 1/n$?

2. Relevant equations

$X_n \overset{a.s.}{\longrightarrow } X$ if $\mathbb{P}( \omega \in \Omega : X_n(\omega) \to X(\omega) \text{ as } n\to \infty) = 1$

3. The attempt at a solution

I don't think I understand this properly. Looking at my attempt I've tried a quick $\epsilon -\delta$ setting $\epsilon = 2/N$ and having $|1/n| < \epsilon$ for $n>N$

I don't think this is what it's asking. Can I say that $X(\omega) = 0$ "clearly" and then that $\mathbb{P}( \omega \in \Omega : |X_n(\omega) - X(\omega)| > \epsilon \text{ i.o. }) = 0$ ?

Where i.o. means infinitely often.