On the difference between consecutive primes Let $(p_n)$ be the sequence of prime numbers and $g_n = p_{n+1} - p_n$  
Question: Is it known that $g_n \le n$?  
Remark: it's known that $g_n < p_n^{\theta}$ with $\theta = 0.525$ for $n$ sufficiently large (see here), and that $p_n < n(\ln n + \ln\ln n )$ for $n \ge 6$ (see here).
   It follows that $g_n < (n(\ln n + \ln\ln n))^{\theta}$  for $n$ sufficiently large.
But $(n(\ln n + \ln\ln n))^{\theta}<n$,  for $n \ge 2$.
Conclusion: it's known that  $g_n \le n$ for $n$ sufficiently large. Is it known for all $n$? 

This should be really easy to answer using Pierre Dusart's explicit estimates on prime-related functions (and probably the older Rosser-Schoenfeld inequalities).  For instance, Proposition 6.8 in "Estimates of some functions on primes without R.H." states that for $x \ge 396738$ there is always a prime in the interval $(x, x + x/(25\ln^2 x)]$.
Since that gap is significantly smaller than $x/\ln x$ and quite explicit, this is certainly less than $\pi(x)$, establishing the inequality for large $n$.  Combined with Galc127's comment about verifying small $n$, that should cover all cases.
