Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 201 / 3 / b

The Poisson central limit theorem, or the ordinary central limit theorem applied to the $X_i$, gives $Y_n\stackrel{d}{\to }Y$ for a standard normal random variable $Y$. The negative-part map $x\mapsto x^{-}:=\max (-x,0)$ is continuous, so the continuous mapping theorem gives