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CIIM 2014 Problem 6

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August 9, 2016
CIIMCIIM 2014undergraduate

Problem Statement

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