How can subscripts converge to $\infty$ almost surely? Here is the question:
Give an example of random variables $X_n \in \{0,1\}$, $X_n \rightarrow 0$ in probability, $N(n) \uparrow \infty$ almost surely and $X_{N(n)} \rightarrow 1$ almost surely.
Here is MY question: 
Is $N(n)$ a random variable? Do we in otherwords have an increasing sequence of random variables: $N(1), N(2), \ldots$ which converge to $\infty$ almost surely?  

Yes, here $N(n)$ is a random variable. It is supposed to converge to $\infty$ in probability, that is, for each $A\gt 0$, $\mathbb P\{|N(n)|\leqslant A\}\to 0$ as $n\to \infty$.
In order to solve the problem, we can take $X_n:=\chi_{A_n}$, where $\mathbb P(A_n)\to 0$ and $\limsup_nA_n=\Omega$.
