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

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 $\sequence {x_n}$ is convergent such that:
 * $\displaystyle \lim_{n \mathop \to \infty} x_n = a$

Proof
First some lemmata:

Lemma 2
From Lemma 1 and Lemma 2 we have that:
 * $0 < a < x_n < b$

for all $n \in \N_{>0}$.

Then:

Hence:
 * $0 < a < x_{n + 1} < x_n < b$

Thus $\sequence {x_n}$ is decreasing and bounded below by $a$.

Hence by the Monotone Convergence Theorem (Real Analysis), $\sequence {x_n}$ converges to its infimum.

It remains to be shown that $\map \inf {x_n} = a$.

Suppose that:


 * $\displaystyle \lim_{n \mathop \to \infty} x_n = l$

Thus:

But $b$ cannot be the infimum of $\sequence {x_n}$ because it is not a lower bound.

Hence:
 * $\displaystyle \lim_{n \mathop \to \infty} x_n = a$

and the result follows.