# Convergent Real Sequence/Examples/x (n+1) = x n^2 + k/Lemma 1

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## Contents

## Example of Convergent Real Sequence

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

- $x_n = \begin {cases} h & : n = 1 \\ {x_{n - 1} }^2 + k & : n > 1 \end {cases}$

where:

- $0 < k < \dfrac 1 4$

- $a < h < b$, where $a$ and $b$ are the roots of the quadratic equation $x^2 - x + k = 0$.

Then:

- $\forall n \in \N_{>0}: a < x_n < b$

## Proof

The proof proceeds by induction.

For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:

- $a < x_n < b$

### Basis for the Induction

$\map P 1$ is the case:

- $a < x_1 < b$

By assertion:

- $a < h < b$

and:

- $x_1 = h$

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:

- $a < x_k < b$

from which it is to be shown that:

- $a < x_{k + 1} < b$

### Induction Step

This is the induction step:

\(\displaystyle x_{k + 1} - a\) | \(=\) | \(\displaystyle {x_k}^2 + k - a\) | Definition of $x_{k + 1}$ | ||||||||||

\(\displaystyle \) | \(>\) | \(\displaystyle a^2 - a + k\) | as $x_k > a$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 0\) | Definition of $a$ | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle x_{k + 1}\) | \(>\) | \(\displaystyle a\) |

Similarly:

\(\displaystyle x_{k + 1} - b\) | \(=\) | \(\displaystyle {x_k}^2 + k - b\) | Definition of $x_{k + 1}$ | ||||||||||

\(\displaystyle \) | \(<\) | \(\displaystyle b^2 - b + k\) | as $x_k < b$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 0\) | Definition of $b$ | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle x_{k + 1}\) | \(<\) | \(\displaystyle b\) |

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

Therefore:

- $\forall n \in \N_{>0}: a < x_n < b$

$\blacksquare$

## Sources

- 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 5$: Subsequences: Exercise $\S 5.7 \ (2)$