In George Casella textbook theorem 5.5.24 Delta Method, it said let $Y_n$ to be a sequence of random variables that satisfies $\sqrt{n}(Y_n-\theta) \rightarrow n(0, \delta^2)$ in distribution. Then we will have a conclusion.
However, from another tutorial material, it said the condition is: let $W_n$ to be a sequence of random variables (same as Casella). If $W_n \sim AN(a, b_n)$, where $b_n \rightarrow 0$ and AN means asymptotically normal. In other words, it said $\frac{W_n-a}{\sqrt{b_n}}\sim N(0,1) $ as $n \rightarrow \infty$.
I can understand that here $W_n$ denotes $Y_n$, $a$ denotes $\theta$, $b_n$ denotes $\delta^2$. But the difference is that Casella has one more $\sqrt{n}$. It really confuses me A LOT. Can anyone help me?