Proving limit of log(log p_n) divided by log n

[AxiomOfChoice]

Let $p_n$ be the nth prime number. Can someone help me figure out how to show that

$$\lim_{n\to \infty} \frac{\log (\log p_n)}{\log n} = 0.$$

You're allowed to assume that

$$\lim_{n\to \infty} \frac{p_n}{n \log p_n} = 1.$$

I'm quite confident what I want to show is true, but it's hard to figure out how to do it because $p_n > n$ for every n. Thanks!

 

[CRGreathouse]

You have that p_n = (1 + o(1))(n log n). Take the log of both sides and rewrite as a limit.