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

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## 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 1

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

$\Box$

### Lemma 2

Consider the subsequences $\sequence {x_{2 n} }$ and $\sequence {x_{2 n - 1} }$.

One of them is strictly increasing and the other is strictly decreasing.

$\Box$

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:

\(\displaystyle l\) | \(=\) | \(\displaystyle \dfrac k {1 + m}\) | |||||||||||

\(\displaystyle m\) | \(=\) | \(\displaystyle \dfrac k {1 + l}\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle l + l m\) | \(=\) | \(\displaystyle k\) | ||||||||||

\(\displaystyle m + l m\) | \(=\) | \(\displaystyle k\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle l\) | \(=\) | \(\displaystyle m\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle l^2 + l\) | \(=\) | \(\displaystyle k\) |

Hence the result.

$\blacksquare$

## Sources

- 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 5$: Subsequences: Exercise $\S 5.7 \ (3)$