This is motivated by this answer to another math.SE question.
One can directly apply the prime number theorem to show that if $p_n$ is the sequence of prime numbers, the limit $\lim_{n\to\infty}p_{n+1}/p_n$ exists (and equals $1$). I'm wondering whether one can show existence of this limit without appealing to the prime number theory, and I'd be especially happy if there's an elegant elementary proof.
Given that my current guess is that such a proof is not too likely to exist, I'd also be happy to know what the core obstacles are.
Edit: I guess I'm really vague about what 'elementary' and 'elegant' mean. I guess it's one of those things that you know when you see them. One particular part of it is that I hope it won't involve too much calculation though.
Best Answer
In An Elementary Proof of the Prime Number Theorem Selberg says that the original elementary proof of Prime Number Theorem uses the following Erdos's result. For any $\delta>0$ there exists an $x_0$ and a constant $K(\delta)>0$ such that for all $x>x_0$ there are more than $K(\delta)\frac{x}{\log x}$ primes between $x$ and $x(1+\delta)$. This result is elementary, does not use prime number theorem(obviously as this result is used to give an elementary proof of prime number theorem) and implies $\lim_{n\rightarrow\infty}\frac{p_{n+1}}{p_n}=1$.