# Weierstrass Product Inequality

## Theorem

For $n \ge 1$:

$\ds \prod_{i \mathop = 1}^n \paren {1 - a_i} \ge 1 - \sum_{i \mathop = 1}^n a_i$

where all of $a_i$ are in the closed interval $\closedint 0 1$.

## Proof

For $n = 1$ we have:

$1 - a_1 \ge 1 - a_1$

which is clearly true.

Suppose the proposition is true for $n = k$, that is:

$\ds \prod_{i \mathop = 1}^k \paren {1 - a_i} \ge 1 - \sum_{i \mathop = 1}^k a_i$

Then:

 $\ds \prod_{i \mathop = 1}^{k + 1} \paren {1 - a_i}$ $=$ $\ds \paren {1 - a_{k + 1} } \prod_{i \mathop = 1}^k \paren {1 - a_i}$ $\ds$ $\ge$ $\ds \paren {1 - a_{k + 1} } \paren {1 - \sum_{i \mathop = 1}^k a_i}$ as $1 - a_{k + 1} \ge 0$ $\ds$ $=$ $\ds 1 - a_{k + 1} - \sum_{i \mathop = 1}^k a_i + a_{k + 1} \sum_{i \mathop = 1}^k a_i$ $\ds$ $\ge$ $\ds 1 - a_{k + 1} - \sum_{i \mathop = 1}^k a_i$ as $a_i \ge 0$ $\ds$ $=$ $\ds 1 - \sum_{i \mathop = 1}^{k + 1} a_i$

Thus, by Principle of Mathematical Induction, the proof is complete.

$\blacksquare$

## Source of Name

This entry was named for Karl Theodor Wilhelm Weierstrass.