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:
\(\ds x_{k + 1} - a\) | \(=\) | \(\ds {x_k}^2 + k - a\) | Definition of $x_{k + 1}$ | |||||||||||
\(\ds \) | \(>\) | \(\ds a^2 - a + k\) | as $x_k > a$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 0\) | Definition of $a$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds x_{k + 1}\) | \(>\) | \(\ds a\) |
Similarly:
\(\ds x_{k + 1} - b\) | \(=\) | \(\ds {x_k}^2 + k - b\) | Definition of $x_{k + 1}$ | |||||||||||
\(\ds \) | \(<\) | \(\ds b^2 - b + k\) | as $x_k < b$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 0\) | Definition of $b$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds x_{k + 1}\) | \(<\) | \(\ds b\) |
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 $\S 5.7 \ (2)$