Hero's Method

Theorem
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 $x_n \to \sqrt a$ as $n \to \infty$.

Lemmata
First we have the following lemmata: