Not a very well known inequality is $$p_{n+1}-p_n\leq n$$ where $p_i$ is the $i^{th}$ prime number.
I know this can be proven using the following inequality: (B. Rosser, L. Schoenfeld) $$\forall x\geq 67 \text{ we have }\frac{x}{\log x-\frac{1}{2}}<\pi(x)$$
and proposition $6.8$ of this article of P. Dussart.
However, I am interested if anyone knows a simpler proof. In particular, one that does not use approximations of $\psi(x)$ and $\vartheta(x)$ presented in the article I linked above.
Thank you!