How do you prove that $\displaystyle\sum^{n+1}\_{j=1}\left|\cos\left(j\cdot x\right)\right|\geqslant \dfrac{n}{4}$, where $x\in\mathbb{R}$?

I tried mathematical induction, but it doesn't work. I also have some ideas on complex number solution, such as setting $z=\cos\left(x\right)+i\sin\left(x\right)$ and working on $Re\left(z\right)$, but neither does that work.