MathLabs

Problem 6

Given any real number a>1, construct a bounded infinite sequence x0,x1,x2,…x_0,x_1,x_2,\ldots such that ∣xn−xm∣ ∣n−m∣a≥1|x_n-x_m|\,|n-m|^a\ge1 for every pair of distinct n,m.
Step 2 of 4: Define the sequence from binary digits
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n=∑i≥0bi2i,xn=1c∑bi=1tin=\sum_{i\ge0}b_i2^i,\qquad x_n=\frac1c\sum_{b_i=1}t^i
Detailed analysis

Write each nonnegative integer n in binary, with bits b_i in {0,1}, and define x_n as the weighted sum of t^i over the positions whose bit is 1, divided by c. All terms are nonnegative and x_n is at most (sum of all t^i)/c=1/(1−2t), so the sequence is bounded.