In my old high school notebook (20 years ago), the following inequality appears with its proof:


$$1+\cos x + \frac{1}{2}\cos 2x + \cdots + \frac{1}{n}\cos nx \geq 0$$
for any real $x$ and positive integer $n$.


I am not the one that created this inequality. So the my question is where references for this inequality can be found.



Need a reference for a trigonometric inequality
According to the last sentence on page 16 of [this paper](https://www.dcce.ibilce.unesp.br/%7Edimitrov/papers/main.pdf), this inequality was proved by


W. H. Young, On certain series of Fourier, Proc. London Math. Soc. (2) 11
(1912), 357–366.


