Law of large numbers for inverse of sample mean

[x100e]

Homework Statement

If $$\overline{X}_n$$ converges to $$\mu$$, does $$\frac{1}{\overline{X}_n}$$ converge to $$\frac{1}{\mu}$$?

Homework Equations

http://mathworld.wolfram.com/WeakLawofLargeNumbers.html

The Attempt at a Solution

$$\frac{1}{\overline{X}_n} = \frac{n}{X_1 + \cdots + X_n}$$

$$E\left(\frac{1}{\overline{X}_n}\right) = E\left(\frac{n}{X_1 + \cdots + X_n}\right)$$

$$E\left(\frac{1}{\overline{X}_n}\right) = n E\left(\frac{1}{X_1 + \cdots + X_n}\right)$$

The RHS does not equal $$1/{\mu}$$ unless $$E\left(\frac{1}{X_1 + \cdots + X_n}\right) = \frac{1}{n \mu}$$ but how can I show that?

I know $$E\left( \frac{1}{\overline{X}_n} \right) \neq \frac{1}{\mu}$$ but I'm not sure how to apply that here.

 

[statdad]

Note that as long as $$x \ne 0$$ the function $$x \to \frac 1 x$$ is continuous.