In mathematics, the Weierstrass product inequality states that for any real numbers 0 ≤ x1, ..., xn ≤ 1 we have
and similarly, for 0 ≤ x1, ..., xn,
where
The inequality is named after the German mathematician Karl Weierstrass.
Proof
The inequality with the subtractions can be proven easily via mathematical induction. The one with the additions is proven identically. We can choose as the base case and see that for this value of we get
which is indeed true. Assuming now that the inequality holds for all natural numbers up to , for we have:
which concludes the proof.
References