A simple Central Limit Theorem question

Question

Let $\{X_1,X_2,\ldots\}$ be random variables with mean $\mu$ and variance $\sigma^2$ both finite.

I know that the CLT gives

$$\frac{\sqrt{n}}{\sigma}\left(\frac{1}{n}\sum_{i=1}^nX_i - \mu\right)\overset{d}{\rightarrow}N(0,1)$$

But is it true that

$$\frac{\sqrt{n}}{\sigma}\left(\frac{1}{n}\sum_{i=1}^nX_i\right)\overset{d}{\rightarrow}N(\mu,1)$$

It seams right, but it cant prove it.

Answer 1

A simple rearrangement of terms proposed by @Glen_b shows that my intuition was wrong. Also a simple way to see why this is not true is to suppose that $X_i\sim N(\mu,\sigma^2)$. Then all we have to do is note that

$$\frac{\sqrt{n}}{\sigma}\left(\frac{1}{n}\sum_{i=1}^n{X_i}\right) \sim N\left(\frac{\sqrt{n}}{\sigma}\mu,1\right)$$

and that would never converge to $N(0,1)$.