How to show that if we have a iid $X_1,X_2,\dots,X_n$ of exponential distribution with parametr $\lambda$ then the estimator of maximum likelihood is consistency. I know that this estimator equals $$\hat{\lambda}_n=\frac{1}{\sum_{i=1}^n X_i}$$ $$P(|\hat{\lambda}_n−λ|>ε)→0 $$ $$P(|\frac{1}{\sum_{i=1}^n X_i}-\lambda|> \epsilon) \rightarrow 0$$ any idea?
$EX_1=\frac{1}{\lambda}$ $$P(|\frac{1}{\hat{\lambda}_n}-\frac{1}{\lambda}|> \epsilon) \rightarrow 0$$ but why we can take inverse of this statistics? $$P(|\frac{1}{\hat{\lambda}_n}-\frac{1}{\lambda}|> \epsilon) \rightarrow 0=P(|\hat{\lambda}_n−λ|>ε)→0 $$ Is it the same?