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

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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