APMO 2018 Problem 2

Let $f(x)$ and $g(x)$ be given by

$$f(x)=\frac{1}{x}+\frac{1}{x-2}+\frac{1}{x-4}+\cdots+\frac{1}{x-2018}$$

and

$$g(x)=\frac{1}{x-1}+\frac{1}{x-3}+\frac{1}{x-5}+\cdots+\frac{1}{x-2017} .$$

Prove that

$$|f(x)-g(x)|>2$$

for any non-integer real number $x$ satisfying $0<x<2018$.