Weierstrass product inequality

In mathematics, the Weierstrass product inequality states that for any real numbers 0 ≤ x₁, ..., x_n ≤ 1 we have

${\displaystyle (1-x_{1})(1-x_{2})(1-x_{3})(1-x_{4})....(1-x_{n})\geq 1-S_{n},}$

and similarly, for 0 ≤ x₁, ..., x_n,^{[1][2]: 210}

${\displaystyle (1+x_{1})(1+x_{2})(1+x_{3})(1+x_{4})....(1+x_{n})\geq 1+S_{n},}$

where ${\displaystyle S_{n}=x_{1}+x_{2}+x_{3}+x_{4}+....+x_{n}.}$

The inequality is named after the German mathematician Karl Weierstrass.