Convergence in probability: The inverse of the simple mean

Question

I have a question on convergence: I have to prove that $\frac{n}{U_{n}} \longrightarrow 1$ in probability, where $U_{n}=\sum X_{i}$, $X_{i}\sim \mathrm{Exp}(1)$ and because of this, $U_{n}\sim \mathrm{Gamma}(n,1)$.

This problem had two parts, the other part was to prove that $\frac{U_{n}}{n}\longrightarrow 1$ in probability which I proved invoking the weak law for big numbers. But this, I have no clue how to prove it.

Thanks so much for your help! :)

Answer 1

I think you can use the continuous mapping theorem on the other part of the question.