Question 145
Let $f(x)$ and $g(x)$ be given by
$$
f(x) = \frac{1}{x} + \frac{1}{x-2} + \dots + \frac{1}{x-2018}
$$
and
$$
g(x) = \frac{1}{x-1} + \frac{1}{x-3} + \dots + \frac{1}{x-2017}.
$$
Prove that
$$
|f(x) - g(x)| \gt 2
$$
for any non-integer real number $x$ satisfying $0 \lt x \lt 2018$.
$$
f(x) = \frac{1}{x} + \frac{1}{x-2} + \dots + \frac{1}{x-2018}
$$
and
$$
g(x) = \frac{1}{x-1} + \frac{1}{x-3} + \dots + \frac{1}{x-2017}.
$$
Prove that
$$
|f(x) - g(x)| \gt 2
$$
for any non-integer real number $x$ satisfying $0 \lt x \lt 2018$.