im working on the following question and i need some help because i have not yet found an idea to tackle this problem.
The question is: Given random variables $X_n$ in a way that $X_n$ converges in distribution against a standard normal distribution. Show that, for a continous function f, the convergence in distribution holds. In other words, show that the following expression convergence in distribution against a standard normal distribution: $\frac{f(X_n)-f(\mu)}{f'(\mu) \sigma_n}$ ($f'(\mu) \neq 0$)
Ive already shown the convergence for $\frac{X_n-\mu}{\sigma_n}$ in another question.
I was thinking about using Slutskys theorem/contionous mapping theorems - since it gives me almost half the answer, and applying the definiton for convergence since this would give me a composition of continous functions. Following this route, i get in trouble explaining the derivative.
Thanks in advance for helping me