Let $X$ be a standard gaussian vector in $\mathbb{R}^n$. I would like to find the limit of $\mathbb{P}(\sqrt n -1 \leq \sqrt{X_1^2 + \cdots X_n^2} \leq \sqrt n + 1)$ as $n \rightarrow \infty$.
This seems like a simple application of the CLT, or maybe the SLLN (the magnitude over $\sqrt n$ goes to $1$?) but I'm having a brain fart.
Edit: Here's something:
I think the inequality can be simplified as $$\frac{1}{\sqrt{2n}} - \sqrt 2 \leq \frac{\sum X_i^2 - n}{\sqrt{2n}} \leq \frac{1}{\sqrt{2n}} + \sqrt 2$$ Since $\mathbb{E}[X_i^2]$ = 1 and Variance is $\sqrt 2$, we may use the central limit theorem to get that this goes to $\mathbb{P}(-\sqrt 2 \leq Z \leq \sqrt 2)$ where $Z \sim N(0,1)$ and this is a constant. Is this correct?