CIIM 2014 Problem 6

a) Let $\{x_n\}$ be a sequence with $x_n \in [0,1]$ for any $n$ . Prove that there exists $C > 0$ such that for every positive integer $r$ , there exists $m \geq 1$ and $n > m + r$ that satisfy $(n-m)|x_n-x_m| \leq C$ . b) Prove that for every $C > 0$ , there exists a sequence $\{x_n\}$ with $x_n \in [0,1]$ for all $n$ and an integer $r$ such that, if $m \geq 1$ and $n > m+r$ , then $(n-m)|x_n-x_m| > C.$