Let $(X_n)_n$ a sequence of i.i.d random variables $\sim \operatorname{Ber}(1/n)$, i. e. $P(X=1)=1/n$ and $P(X=0)=1-1/n$.
My question: Using Borel-Cantelli, I can show that both $\{X_n=1\}_n$ and $\{X_n=0\}_n$ will occur infinetly many times. I think this contradicts the almost sure convergence. But can this really happen? Can both events occur infintely many often? Is this like $\infty+\infty=\infty$?