The folowing is stated and proved in the random graphs book by Luczak, Janson, Rucinski and this is on page 108 in the Giant component section.
I can't understand why the conclusion follows from the last statement in the proof.
Question: We consider a series of branching processes with immediate offspring distribution $X_n \sim \mathrm{Bin}(n,p)$ where $np \to c$ as $n\to \infty$ for some constant $c>1$.
We also consider a random variable $X \sim \mathrm{Poiss}(c)$.
If we denote by $\rho_{X_i}$ the extinction probability.
Show $\rho_{X_i} \to \rho_X$.
Solution: This is the solution given in my textbook, it seems wrong to me and I don't seem to be able to fix it, it is quite important to me as the remainder of the chapter is based on it.
We know that generating functions are $$f_{X_n}(x)=(1-p+xp)^n$$ and
$$f_X(x)=e^{c(x-1)}$$
It is now a standard result that for a fixed $x$ $f_{X_n}(x) \to f_X(x)$ so $f_n$ converge pointwise to $f$.
As $c<1$ we have $f_{X_n}(\rho_{X_n})=\rho_{X_n}$ and $f_{X}(\rho_{X})=\rho_{X}$
Now the book just states that this implies $\rho_{X_n} \to \rho_X$. And I just don't see why.