Let $(X_n)_n$ be a sequence of i.i.d. random variables with $\mathbb{E}X_1 = 1, \operatorname{Var}(X_1) = 3$.
Show that $\sqrt{\sum_{i}^n X_i} - \sqrt{n} \to Y \sim \mathcal{N}(\mu, \sigma^2)$ in distribution for some numbers $\mu, \sigma \in \mathbb{R}$ and determine those numbers.
My attempt:
$$\mathbb{P}\{\sqrt{\sum_{i=1}^n X_i} - \sqrt{n} \leq x\} = \mathbb{P}\{0 \leq \sum X_i \leq (x + \sqrt{n})^2\} $$
$$= \mathbb{P}\{\sum X_i \leq (x+ \sqrt{n})^2\} - \mathbb{P}\{\sum X_i < 0\}$$
and this last probability seems $0$ to me. Because $\sum X_i < 0$ is not possible, or the square root would not be well defined. But I think this is suspicious because it isn't given that $\sum X_i \geq 0$. What do you think?
I want to apply the central limit theorem, but my problem is that the right hand side depends on $n$, so this isn't allowed.
Maybe I have to use some continuity argument to proceed?