Question about the prime number theorem

[AxiomOfChoice]

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

<br /> \lim_{n\to \infty} \frac{\log (\log p_n)}{\log n} = 0.<br />

You're allowed to assume that

<br /> \lim_{n\to \infty} \frac{p_n}{n \log p_n} = 1.<br />

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 &gt; 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.