Suppose that $X_{n}\sim \text{Binomial}(n,\theta)$, where $n=1,2,\ldots$ and $0<\theta<1$. Find a function $g$ such that $\sqrt{n}(g(\frac{1}{n}X_n)-g(\theta))\xrightarrow{D} N(0,1)$ for each value of $\theta\in(0,1)$. Can someone give me hints by using Delta method? I was trying to prove $\sqrt{n}(X_{n}-\theta)\xrightarrow{D} N(0,\sigma^2)$. But I find I only know $\bar{X}_n$ has similar property.
Delta Method Theorem Let $Y_n$ be sequence of random variables that satisfies $\sqrt{n}(Y_n-\theta)\xrightarrow{D}N(0,\sigma^2)$. For a given function $g$ and a specific value of $\theta$, suppose that $g'(\theta)$ exists and is not $0$. Then $$\sqrt{n}[g(Y_n)-g(\theta)]\xrightarrow{D}N(0,\sigma^2[g'(\theta)]^2).$$