Question 153
Denote by $\{x\}$ the fractional part of a real number $x$, that is $\{x\} = x - \lfloor x \rfloor$ where $\lfloor x \rfloor$ is the largest integer not greater than $x$. Prove that
1. For every positive integer $n$, we have $\{n \sqrt{17}\} \gt \frac{1}{2 \sqrt{17} \cdot n}$.
2. The value $\frac{1}{2 \sqrt{17}}$ is the largest constant $c$ such that the inequality $\{n \sqrt{17}\} \gt \frac{c}{n}$ holds for all positive integers $n$.
1. For every positive integer $n$, we have $\{n \sqrt{17}\} \gt \frac{1}{2 \sqrt{17} \cdot n}$.
2. The value $\frac{1}{2 \sqrt{17}}$ is the largest constant $c$ such that the inequality $\{n \sqrt{17}\} \gt \frac{c}{n}$ holds for all positive integers $n$.