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

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:

Similarly:

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

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