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

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 $\sequence {x_n}$ is convergent to the positive root of the quadratic equation:
 * $x^2 + x = k$

Proof
First some lemmata:

Lemma 2
From Lemma 2, We have that both $\sequence {x_{2 n} }$ and $\sequence {x_{2 n - 1} }$ is strictly monotone (one strictly increasing and the other strictly decreasing).

From Lemma 1, they are both bounded above by $k$ and bounded below by $0$.

Hence from the Monotone Convergence Theorem (Real Analysis), they both converge.

Let:


 * $x_{2 n} \to l$ as $n \to \infty$


 * $x_{2 n - 1} \to m$ as $n \to \infty$

Then:

Hence the result.