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A. 624. $\displaystyle a)$ Prove that for every infinite sequence $\displaystyle x_1,x_2,\ldots\in[0,1]$ there exists some $\displaystyle C>0$ such that for every positive integer $\displaystyle r$ there are positive integers $\displaystyle n$, $\displaystyle m$ satisfying $\displaystyle |n-m|\ge r$ and $\displaystyle |x_n-x_m|<\frac{C}{|n-m|}$.

$\displaystyle b)$ Show that for every $\displaystyle C>0$ there exists an infinite sequence $\displaystyle x_1,x_2,\ldots\in[0,1]$ and a positive integer $\displaystyle r$ such that $\displaystyle |x_n-x_m|>\frac{C}{|n-m|}$ holds true for every pair $\displaystyle n$, $\displaystyle m$ of positive integers with $\displaystyle |n-m|\ge r$.

(CIIM6, Costa Rica)

(5 points)

Deadline expired on 10 November 2014.

Statistics on problem A. 624.
 5 students sent a solution. 5 points: Williams Kada. 2 points: 2 students. 0 point: 2 students.

• Problems in Mathematics of KöMaL, October 2014

•  Támogatóink: Morgan Stanley