Central Limit Theorem Fails for Cauchy Distribution

A direct corollary of Central Limit Theorem is: for i.i.d Random Variable set $\{X_i{\}}_{i=1}^n$ with same Mean $\mu$ and Variance ${\sigma }^2$,

$$\frac{{\sum }_{i=1}^n{X}_i}{n}\stackrel{d}{⟶}Z\sim \mathcal{N}\left(\mu ,\frac{{\sigma }^2}{n}\right).$$

However, we know that the sample mean of Cauchy distribution is

$$\frac{{\sum }_{i=1}^n{X}_i}{n}\sim \mathrm{Cauchy}\left(a\right),$$

where $a$ is the parameter of $X_i$. Therefore, CLT fails for Cauchy Distribution.