Two questions regarding applications of the CLT

Question

Suppose $X_1,\ldots,X_n$ are iid random variables with mean $0$ and variance $1$ . By the CLT we know that $\frac{1}{\sqrt{n}}\sum_{i=1}^n X_i$ converges in distribution to a standard normal distribution.

1. Can you infer from the CLT that $\left(\frac{1}{\sqrt{n}}\sum_{i=1}^n X_i \right)^2$ converges in distribution to Chi-Squared with 1 degree of freedom?
2. Can you infer that $\max\left\{ 0,\frac{1}{\sqrt{n}}\sum_{i=1}^n X_i\right\}$ converges in distribution to the distribution of $\max\left\{ 0,Z\right\}$ where $Z\sim\text{Normal}\left(0,1\right)$ ?

I'm led to believe both of these things are true but I can't manage to justify to myself why. Help would be appreciated.

Answer 1

As pointed out by d.k.o., everything holds as a consequence of the continuous mapping theorem, which states the following: if $\left(Z_n\right)_{n\geqslant 1}$ is a sequence of (real-valued) random variables which converges in distribution to $Z$, then for all continuous function $g\colon\mathbb R\to\mathbb R$, the sequence $\left(g\left(Z_n\right)\right)_{n\geqslant 1}$ converges in distribution to $g\left(Z\right)$.

1. Apply this to $g\colon x\mapsto x^2$.
2. Apply this to $g\colon x\mapsto \max\left\{x,0\right\}$.