, 2014

$a)$ Prove that for every infinite sequence $x_1,x_2,\ldots\in[0,1]$ there exists some $C>0$ such that for every positive integer $r$ there are positive integers $n$, $m$ satisfying $|n-m|\ge r$ and $|x_n-x_m|<\frac{C}{|n-m|}$.
$b)$ Show that for every $C>0$ there exists an infinite sequence $x_1,x_2,\ldots\in[0,1]$ and a positive integer $r$ such that $|x_n-x_m|>\frac{C}{|n-m|}$ holds true for every pair $n$, $m$ of positive integers with $|n-m|\ge r$.
(CIIM6, Costa Rica)
(5 pont)