Bounding a sum of inverses of absolute values

I am studying the proof in Zygmund's Trigonometric Series of the existence of an integrable function whose Fourier series diverges everywhere. I am stuck in a part where it is said that if $n \in \mathbb{N}$ and $l$ is an odd natural number with $0 < l < 2n$, then$$-\frac{2n+1}{2n}\sum_{i=1}^n \frac{1}{|l-2i|} \geq -C\log(n)$$for some constant $C > 0$ that may depend on $n$. I have been trying to prove that$$\sum_{i=1}^n \frac{1}{|l-2i|} \leq C\log(n),$$but I haven't even managed to do it for the simplest case ($l = 1$).