Convergent Real Sequence/Examples/Term is Geometric Mean of Preceding Two Terms

Example of Convergent Real Sequence
Let $a, b \in \R_{>0}$ be (strictly) positive real numbers such that $a \le b$.

Let $a, b \in \R_{>0}$ be such that $a \le p \le q \le b$.

Let $\sequence {x_n}_{n \mathop \in \N_{>0} }$ be the sequence in $\R$ defined as:
 * $x_n = \begin {cases} p & : n = 1 \\ q & : n = 2 \\ \sqrt{x_{n - 1} x_{n - 2} } & : n > 2 \end {cases}$

That is, beyond the first $2$ terms, each term is the geometric mean of the previous $2$ terms.

Then $\sequence {x_n}$ converges.

Proof
First note that:
 * $a \le x_i \le b$ for $i \in \set {1, 2}$.

It can be shown by Proof by Mathematical Induction that:
 * $\forall n \in \N_{>0}: a \le x_n \le b$

Then:

and:

That is:


 * $\dfrac a b \le \dfrac {x_{n + 1} } {x_n} \le \dfrac b a$

Then we have:

Let $n > m$ be arbitrary. Then:

Therefore $\sequence {x_n}$ is a Cauchy sequence and hence converges.