Does negative part of a standardized random variable converge to negative part of a $\mathcal{N}(0,1)$?

Question

I know how to prove that any standardized random variable converge in distribution to a $\mathcal{N}(0,1)$, I was wondering if even $f((S_n-n)/ \sqrt{n}))$ converge to $f(\mathcal{N}(0,1))$, in particular here I have that $S_n$ is a sum of Poisson random variable (with parameter 1), and $f$ is negative part (but I'd like a more general result if there is)

$((S_n-n)\ \sqrt{n})^- \to \mathcal{N^-}(0,1)$ (in distribution)

Answer 1

If $(Y_n)_{n\geqslant 1}$ is a sequence of real valued random variables which converge in distribution to $Y$ and $g\colon\mathbf R\to\mathbf R$ is a continuous function, then $(g(Y_n))_{n\geqslant 1}$ converges to $g(Y)$ in distribution.