Let $X_i \sim N(\mu,\sigma^2)$, I want to find out if $$\frac{1}{\frac{1}{n}\sum_{i=1}^{n}\frac{1}{X_{i}}}$$ is a consistent estimator for $\mu$, or not. It's easy to show, using $LLN$, that $\overline{X}$ is a consistent estimator for $\mu$, and I figure I'm supposed do some manipulation to turn this into that, but so far I haven't been able to do so.
How do I proceed?