Hero's Method/Lemma 1
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:
- $\forall n \in \N_{>0}: x_n > 0$
Proof
The proof proceeds by induction.
For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
- $x_n > 0$
Basis for the Induction
$\map P 1$ is the case:
- $x_1 > 0$
which is assumed.
Thus $\map P 1$ is seen to hold.
This is the basis for the induction.
Induction Hypothesis
Now it needs to be shown that if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.
So this is the induction hypothesis:
- $x_k > 0$
from which it is to be shown that:
- $x_{k + 1} > 0$
Induction Step
This is the induction step:
We have that:
- $x_{k + 1} = \dfrac {x_k + \dfrac a {x_k} } 2$
But as $x_k > 0$ and $a > 0$, it follows that:
- $\dfrac a {x_k} > 0$
Then as $x_k > 0$ and $\dfrac a {x_k} > 0$, it follows that:
- $\dfrac 1 2 \paren {x_k + \dfrac a {x_k} } > 0$
So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.
Therefore:
- $\forall n \in \N_{>0}: x_n > 0$
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 5$: Subsequences: $\S 5.5$: Example