I'm reading through Random Graphs by Janson, Luczak, Rucinski. There is an argument (proposition 1.15) that uses the result $$\Pr(X_n<M_n)\to 1/2$$ as $n\to\infty$, where $X_n\sim \mathrm{Bin}(N_n,M_n/N_n)$, and $N_n,M_n$ vary with $n$ such that $0\le M_n\le N_n$ and $M_n(N_n-M_n)/N_n\to \infty$.
The book appeals to the central limit theorem for this result, but I'm having trouble seeing why it follows. $X_n$ is a sum of $\mathrm{Be}(p_n:=M_n/N_n)$ random variables, but as far as I know the CLT needs $p_n$ to be constant. After some googling, the Berry Esseen theorem seems to give a bound that works, but I don't think it's what the authors intended. Is there a simple argument I'm missing?
(really, it suffices to prove that $\Pr(X_n<M_n)$ doesn't converge to 0 or 1, but I can't even see why that's true.)
Indeed CLT holds and yields the result, as can be seen by considering $Y_n=(X_n-M_n)/s_n$ where $s_n^2=M_n(N_n-M_n)/N_n$ is the variance of $X_n$.
To see this, note that the hypothesis that $s_n\to\infty$ is exactly what is needed to guarantee, by an expansion of up to second order around zero, that $E[\mathrm e^{\mathrm itY_n}]\to\mathrm e^{-t^2/2}$ for every real $t$. Thus, $Y_n$ converges in distribution to a standard normal random variable $Y$, in particular $$ P[X_n\lt M_n]=P[Y_n\lt0]\to P[Y\lt0]=\tfrac12. $$