Let b > a > 0 {\displaystyle {}b>a>0} be positive real numbers. We define recursively two sequences ( x n ) n ∈ N {\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }} and ( y n ) n ∈ N {\displaystyle {}{\left(y_{n}\right)}_{n\in \mathbb {N} }} such that x 0 = a {\displaystyle {}x_{0}=a} , y 0 = b {\displaystyle {}y_{0}=b} , and that
Prove that [ x n , y n ] {\displaystyle {}[x_{n},y_{n}]} is a sequence of nested intervals.