Define $s_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime number, now how to show that $$\lim_{n \rightarrow \infty} \inf \frac{s_n}{\log n} \leq 1$$ I used the result from the prime number theorem: $p_n \sim n \log n$, but have a difficult time dealing with the $\inf$ part.
How to prove this inequality using prime number theorem
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prime-numbers
analytic-number-theory