# Intervallschachtelung/Arithmetisches und geometrisches Mittel/Aufgabe/en

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Let ${\displaystyle {}b>a>0}$ be positive real numbers. We define recursively two sequences ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ and ${\displaystyle {}{\left(y_{n}\right)}_{n\in \mathbb {N} }}$ such that ${\displaystyle {}x_{0}=a}$, ${\displaystyle {}y_{0}=b}$ and that

${\displaystyle x_{n+1}={\text{ geometric mean of }}x_{n}{\text{ and }}y_{n},}$
${\displaystyle y_{n+1}={\text{ arithmetic mean of }}x_{n}{\text{ and }}y_{n}.}$

Prove that ${\displaystyle {}[x_{n},y_{n}]}$ is a sequence of nested intervals.