MathDB

Problem 6

Part of 2014 CIIM

Problems(1)

CIIM 2014 Problem 6

Source:

8/9/2016
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.
CIIMCIIM 2014undergraduate