Problem 6

Given any real number a>1, construct a bounded infinite sequence

x_0,x_1,x_2,\ldots

such that

|x_n-x_m|\,|n-m|^a\ge1

for every pair of distinct n,m.

Step 4 of 4: Bound the tail and finish

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c|x_n-x_m|>ct^k,\qquad |x_n-x_m|\,|n-m|^a>ct^k2^{ak}=1

Detailed analysis

At the first differing bit, the contribution to c times the difference has magnitude t^k; the remaining tail is bounded below by minus the sum from i=k+1 onward, namely minus t^(k+1)/(1−t). Thus c|x_n−x_m| is greater than t^k(1−t/(1−t))=ct^k. Multiplying by |n−m|^a≥2^(ak) gives a product greater than t^k2^(ak)=1, as required.