Consider a random sample from exponential distribution with mean $\frac{1}{\theta}$. I have to prove that $nX_{(1)}$ is not consistent for $\frac{1}{\theta}$ . A sufficient condition for consistency is
$$\lim_{n \to \infty}E(\hat\theta)=\frac{1}{\theta}\quad ;\quad \lim_{n \to \infty}\operatorname{Var}(\hat\theta)=0$$
Since $X_{(1)}$ follow $\exp(n\theta)$, the second condition, was seen to be violated as its variance $\frac{1}{\theta^{2}}$ doesn't tend to zero. Is this enough to prove that $nX_{(1)}$ is inconsistent?
Can i say that since $X_{(1)}$ converge in probability to $\frac{1}{n\theta}$, $nX_{(1)}$ converge in probability to $\frac{1}{\theta}$. But this does not prove what i want.
Im confused about the right method to be used.