Prove that 1 - 2x/(x^2+1)>=0

We'll multiply 1 by (x^2 + 1) and we'll
get:

(x^2 + 1 - 2x)/(x^2 + 1)
>=0

A fraction is positive when the numerator and
denominator are both positive or both negative.

The
numerator of the fraction is the perfect square (x-1)^2 that is positive for any real x,
except x = 1, for the numerator is cancelling out.

The
denominator is always positive, for any real value of
x.

Therefore, the inequality 1 - 2x/(x^2+1) =
(x-1)^2/(x^2 + 1) >=0 is verified for any real
x.