Convergent Real Sequence/Examples/x (n+1) = k over 1 + x n/Lemma 1

Example of Convergent Real Sequence
Let $h, k \in \R_{>0}$.

Let $\sequence {x_n}$ be the real sequence defined as:


 * $x_n = \begin {cases} h & : n = 1 \\ \dfrac k {1 + x_{n - 1} } & : n > 1 \end {cases}$

Then:
 * $\forall n \in \N_{>1}: k > x_n > 0$

Proof
The proof proceeds by induction.

For all $n \in \Z_{>1}$, let $\map P n$ be the proposition:
 * $k > x_n > 0$

Basis for the Induction
$\map P 2$ is the case:
 * $k > x_2 > 0$

We have:

Also, as $k > 0$ and $x_1 > 0$ we have that:
 * $\dfrac k {1 + x_1} > 0$

Thus $\map P 2$ is seen to hold.

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that if $\map P r$ is true, where $r \ge 2$, then it logically follows that $\map P {r + 1}$ is true.

So this is the induction hypothesis:
 * $k > x_r > 0$

from which it is to be shown that:
 * $k > x_{r + 1} > 0$

Induction Step
This is the induction step:

Also, as $k > 0$ and $x_r > 0$ we have that:
 * $\dfrac k {1 + x_r} > 0$

So $\map P r \implies \map P {r + 1}$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\forall n \in \N_{>1}: k > x_n > 0$