MathDB

Sequence + Inequality Problem

Source: 2018 Thailand October Camp 3.1

February 19, 2022

inequalitiesSequencealgebra

Problem Statement

Let

\{x_i\}^{\infty}_{i=1}

and

\{y_i\}^{\infty}_{i=1}

be sequences of real numbers such that

x_1=y_1=\sqrt{3}

, x_{n+1}=x_n+\sqrt{1+x_n^2} \text{and} y_{n+1}=\frac{y_n}{1+\sqrt{1+y_n^2}} for all

n\geq 1

. Prove that

2<x_ny_n<3

for all

n>1