MathLabs

Problem 6

Let a1,…,ana_1,\ldots,a_n be positive numbers and let 0<q<10<q<1. Find real numbers b1,…,bnb_1,\ldots,b_n such that (1) ak<bka_k<b_k for every kk; (2) q<bk+1/bk<1/qq<b_{k+1}/b_k<1/q for k=1,…,n−1k=1,\ldots,n-1; and (3) ∑k=1nbk<1+q1−q∑k=1nak\sum_{k=1}^n b_k<\frac{1+q}{1-q}\sum_{k=1}^n a_k.
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In plain words

Positivity preserves the strict inequality after summing.

∑br<1+q1−q∑ai\sum b_r<\frac{1+q}{1-q}\sum a_i
Detailed analysis

Multiplying the strict inner bounds by ai>0a_i>0 and summing proves condition (3), completing the construction.