Consider $X_1, X_2, ... ,X_n \sim_{i.i.d} Cauchy(\theta), \bar{X} = \frac{1}{n}\sum_{i=1}^n{X_i}$
To prove that it is inconsistant, consider the characteristic function of $X_i$ and $\bar{X}$, which are both $\exp\{i\theta t+|t|\}$. i.e. $\bar{X} \sim Cauchy(\theta)$. Does this prove $\bar{X}$ is inconsistent because consistent estimators' characteristic function should be $e^{ti\theta}$?
I'm just having difficulties relating the moment generating functions and characteristic functions to the definition of consistency.