Assume a Poisson point process in $[0,1]^2$, with intensity $\mu$. For any positive integer $n$, consider the division of the space into $4^n$ disjoint half open squares with side length $2^{-n}$. Let $N_n$ be the number of such half open squares that contain points of the process. Find the almost sure limit of $N_n$.
First, I note that for each such small square, it contains points with Poisson distribution of intensity $\mu4^{-n}$. Hence, the probability that each square contains at least a point is $1-e^{-\mu 4^{-n}}$. Hence, by independence of each squares, I can see that $N_n$ follows a binomial distribution $B(4^n,p_n)$, where $p_n=1-e^{-\mu 4^{-n}}$. I can see that the probability of any small square containing more than a point should go to $0$, so my guess is $N_n$ should converge to some Poisson distribution, but I don't have any idea on how to progress from here.