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 1 of 4: Choose a geometric weight
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t=2−a,c=1−t1−t>0t=2^{-a},\qquad c=1-\frac{t}{1-t}>0
Detailed analysis

Because a>1, t=2 to the power minus a is less than one half, so c=1−t/(1−t) is positive. The constant c will absorb the possible tail cancellation after the first differing binary digit.