Convergent Real Sequence/Examples/x (n+1) = k over 1 + x n/Lemma 1
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Example of Convergent Real Sequence
Let $h, k \in \R_{>0}$.
Let $\sequence {x_n}$ be the real sequence defined as:
- $x_n = \begin {cases} h & : n = 1 \\ \dfrac k {1 + x_{n - 1} } & : n > 1 \end {cases}$
Then:
- $\forall n \in \N_{>1}: k > x_n > 0$
Proof
The proof proceeds by induction.
For all $n \in \Z_{>1}$, let $\map P n$ be the proposition:
- $k > x_n > 0$
Basis for the Induction
$\map P 2$ is the case:
- $k > x_2 > 0$
We have:
\(\ds x_2\) | \(=\) | \(\ds \dfrac k {1 + x_1}\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds \dfrac k 1\) | as $x_1 > 0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds k\) |
Also, as $k > 0$ and $x_1 > 0$ we have that:
- $\dfrac k {1 + x_1} > 0$
Thus $\map P 2$ is seen to hold.
This is the basis for the induction.
Induction Hypothesis
Now it needs to be shown that if $\map P r$ is true, where $r \ge 2$, then it logically follows that $\map P {r + 1}$ is true.
So this is the induction hypothesis:
- $k > x_r > 0$
from which it is to be shown that:
- $k > x_{r + 1} > 0$
Induction Step
This is the induction step:
\(\ds x_{r + 1}\) | \(=\) | \(\ds \dfrac k {1 + x_r}\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds \dfrac k 1\) | Induction Hypothesis: $x_r > 0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds k\) |
Also, as $k > 0$ and $x_r > 0$ we have that:
- $\dfrac k {1 + x_r} > 0$
So $\map P r \implies \map P {r + 1}$ and the result follows by the Principle of Mathematical Induction.
Therefore:
- $\forall n \in \N_{>1}: k > x_n > 0$
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 5$: Subsequences: Exercise $\S 5.7 \ (3)$