Two sequences of positive integers, x_{1}, x_{2}, x_{3}, \ldots and y_{1}, y_{2}, y_{3}, \ldots, are given, such that
y_{n+1} / x_{n+1}>y_{n} / x_{n}
for each n \geq 1. Prove that there are infinitely many values of n such that y_{n}>\sqrt{n}.
Two sequences of positive integers, x_{1}, x_{2}, x_{3}, \ldots and y_{1}, y_{2}, y_{3}, \ldots, are given, such that
for each n \geq 1. Prove that there are infinitely many values of n such that y_{n}>\sqrt{n}.