Quoting from Arthur Engel's "Problem Solving Strategies":

To the part inred, how is this possible? If $\displaystyle x_{n}$ keeps increasing, $\displaystyle y_{n}$ has to decrease in order to keep the product of $\displaystyle x_{n}$ and $\displaystyle y_{n}$ constant at $\displaystyle ab$.

To the part inblue, I don't get this.

To the part ingreen, isn't harmonic and arithmetic mean equal in magnitude always?

Thanks. Also I don't get the inequality which he has generated. What is he trying to show by $\displaystyle x_{n+1} - y_{n+1}$?