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

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## Example of Convergent Real Sequence

Let $a$ and $b$ be the roots of the quadratic equation:

- $(1): \quad x^2 - x + k = 0$

Let:

- $0 < k < \dfrac 1 4$

Then $a$ and $b$ are both strictly positive real numbers.

## Proof

First we investigate the consequences of the condition $k < \dfrac 1 4$.

By Solution to Quadratic Equation with Real Coefficients:

- In order for the quadratic equation $a x^2 + b x + c$ to have real roots, its discriminant $b^2 - 4 a c$ needs to be strictly positive.

The discriminant $D$ of $(1)$ is:

\(\displaystyle D\) | \(=\) | \(\displaystyle \paren {-1}^2 - 4 \times 1 \times k\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 1 - 4 k\) |

Thus:

\(\displaystyle k\) | \(<\) | \(\displaystyle \dfrac 1 4\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle 1 - 4 k\) | \(>\) | \(\displaystyle 0\) |

and so when $k < \dfrac 1 4$, $(1)$ has real roots.

$\Box$

Next we investigate the consequences of the condition $0 < k$.

By Solution to Quadratic Equation:

\(\displaystyle x\) | \(=\) | \(\displaystyle \dfrac {-b \pm \sqrt {b^2 - 4 a c} } {2 a}\) | where $b = -1$, $a = 1$, $c = k$ in $(1)$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac {-\paren {-1} \pm \sqrt {\paren {-1}^2 - 4 \times 1 \times k} } {2 \times 1}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac {1 \pm \sqrt {1 - 4 k} } 2\) |

We have that:

\(\displaystyle 0\) | \(<\) | \(\displaystyle k\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle 1\) | \(>\) | \(\displaystyle 1 - 4 k\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle 1\) | \(>\) | \(\displaystyle +\sqrt {1 - 4 k}\) | ||||||||||

\(\displaystyle -1\) | \(<\) | \(\displaystyle -\sqrt {1 - 4 k}\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \dfrac {1 \pm \sqrt {1 - 4 k} } 2\) | \(>\) | \(\displaystyle 0\) |

That is, when $0 < k$ both roots of $(1)$ are strictly positive.

$\Box$

Hence when $0 < k < \dfrac 1 4$, both roots of $(1)$ are strictly positive real numbers.

$\blacksquare$

## Sources

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