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

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:

Thus:

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

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

By Solution to Quadratic Equation:

We have that:

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

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