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Bounding difference of sum of fractions in an interval

Source: APMO 2018 P2

June 24, 2018

algebraAPMO

Problem Statement

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}

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

.

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