Limiting distribution of a sequence of random variables

Question

$X_1$,...$X_n$,... are iid random variables with mean being $1$ and variance being $1$. Let $S_n=\sum_{i=1}^{n}X_i$. Let $\Phi(\cdot)$ be the cdf of standard normal distribution. What is the limiting distribution of $\sqrt{n}\left[\Phi(S_n/n)-\Phi(1)\right]$? Justify the statement.

Answer 1

By the CLT

$$\sqrt{n}(S_n/n-1)\xrightarrow{d}N(0,1).$$

Then using the delta method,

$$\sqrt{n}(\Phi(S_n/n)-\Phi(1))\xrightarrow{d}N(0,\phi(1)^2).$$