Hero's Method/Proof 1
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 $n \to \infty$.
Proof
First we have the following lemmata:
Lemma 1
- $\forall n \in \N_{>0}: x_n > 0$
$\Box$
Lemma 2
- $\forall n \ge 2: x_n \ge \sqrt a$
$\Box$
Consider $x_n - x_{n + 1}$.
\(\ds x_n - x_{n + 1}\) | \(=\) | \(\ds x_n - \frac {x_n + \dfrac a {x_n} } 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 x_n} \paren {x_n^2 - a}\) | ||||||||||||
\(\ds \) | \(\ge\) | \(\ds 0\) | for $n \ge 2$ | \(\quad\) Lemma 2 |
So, providing we ignore the first term (about which we can state nothing), the sequence $\sequence {x_n}$ is decreasing and bounded below by $\sqrt a$.
Thus by the Monotone Convergence Theorem (Real Analysis), $x_n \to l$ as $n \to \infty$, where $l \ge \sqrt a$.
Now we want to find exactly what that value of $l$ actually is.
By Limit of Subsequence equals Limit of Real Sequence we also have $x_{n + 1} \to l$ as $n \to \infty$.
But $x_{n + 1} = \dfrac {x_n + \dfrac a {x_n} } 2$.
Because $l \ge \sqrt a$ it follows that $l \ne 0$.
So by the Combination Theorem for Sequences:
- $x_{n + 1} = \dfrac {x_n + \dfrac a {x_n} } 2 \to \dfrac {l + \dfrac a l} 2$ as $n \to \infty$
Since a Convergent Real Sequence has Unique Limit, that means:
- $l = \dfrac {l + \dfrac a l} 2$
and so (after some straightforward algebra):
- $l^2 = a$
Thus:
- $l = \pm \sqrt a$
and as $l \ge +\sqrt a$ it follows that:
- $l = +\sqrt a$
Hence the result.
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 5$: Subsequences: $\S 5.5$: Example