Prove or disprove $\lim\limits_{n \to \infty} (p_{n+1} - p_{n})/\sqrt{p_n} = 0$ Can anyone prove or disprove the following statement?

$$ \lim_{n \to \infty} \frac{p_{n+1} - p_{n}}{\sqrt{p_n}} = 0.$$

 A: I don't think anyone knows, although I am looking up stuff just in case. Meanwhile, what people suspect is the Cramer-Granville conjecture,
$$ \lim \sup \frac{p_{n+1}-p_n}{\left( \log p_n \right)^2} = 2 e^{- \gamma} = 1.1229\ldots,   $$
where the logarithm is to base $e = 2.718281828459\ldots$ and $\gamma = 0.5772156649\ldots$ is the Euler-Mascheroni constant. This conjecture, and the Baker result mentioned in the other answer, are in GRANVILLE PDF and WookiePedia. Hmmm, not quite, Granville mentions the earlier $0.535$ result of Baker and Harman. With Pintz they later got it to $0.525.$
This is consistent with a (mostly) stronger conjecture that I made up for no good reason except that it also applies to small numbers,
$$  p_{n+1} \, - \, p_n < \; 3 \; \log^2 \, p_n.   $$ 
For example, $$p_1 = 2,\; \log 2 = 0.693147\ldots, \log^2 \, 2 = (0.693147\ldots)^2 = 0.480453\ldots, \; 3 \,\log^2 \, 2 = 1.441359\ldots,  $$ 
and 
$$  2 + 1.441359\ldots > 3 = p_2.  $$
A: At present, no one can prove or disprove this statement.  It is very famous and still open.  The best unconditional result is Baker-Harman-Pintz:
$$\lim_{n \to \infty} \frac{p_{n+1} - p_{n}}{p_n^{0.525}} < \infty.$$
Quite likely it can be shown that this limit is exactly $0$, but I haven't read enough of the paper to be certain.
A: This is not known even under the Riemann hypothesis, which gives only
$$
\frac{p_{n+1}-p_n}{\sqrt{p_n}\log p_n}<\infty.
$$
