Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 14 / 4 / b / Solution

The factorial-moment argument above gives $Y\Rightarrow \mathrm{Poisson}(\lambda )$ with $\lambda =e^{-c}$. Because $Y$ is integer-valued, apply convergence in distribution at the continuity points $3/2$ and $5/2$ and subtract. This proves