Hero's Method/Lemma 2

Lemma for Hero's Method
Let $a \in \R$ be a real number such that $a > 0$.

Let $x_1 \in \R$ be a real number such that $x_1 > 0$.

Let $\sequence {x_n}$ be the sequence in $\R$ defined recursively by:


 * $\forall n \in \N_{>0}: x_{n + 1} = \dfrac {x_n + \dfrac a {x_n} } 2$

Then:
 * $\forall n \ge 2: x_n \ge \sqrt a$

Lemma 1
We have:

This is a quadratic equation in $x_n$.

We know that this equation must have a real solution with respect to $x_n$, because $x_n$ has been explicitly constructed by the iterative process.

Thus its discriminant is $b^2 - 4 a c \ge 0$, where:


 * $a = 1$
 * $b = -2 x_{n + 1}$
 * $c = a$

Thus $x_{n + 1}^2 \ge a$.

From Lemma 1:
 * $x_{n + 1} > 0$

It follows that:
 * $\forall n \ge 1: x_{n + 1} \ge \sqrt a$ for $n \ge 1$

Thus:
 * $\forall n \ge 2: x_n \ge \sqrt a$ for $n \ge 2$