Nested intervals/Arithmetic and geometric mean/Exercise

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

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

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

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

Create a solution