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Spacy Colors Lemma (Posted on 2014-12-06) Difficulty: 4 of 5

Let { ak } be any sequence of real numbers that satisfies
   ak ≥ ak+1 > 0 for all k≥1.
Let { rk } be the sequence of real numbers that satisfies
   rk = ak                        for k = 1,

      = ak*√[ 1 - tk*tk ]          for k > 1,

   where tk = ak/(2*rk-1).
Clearly, for all k > 1, rk is defined and greater than zero
if tk∈(0,1).

Prove or disprove that tk∈(0,1) for all k > 1.

See The Solution Submitted by Bractals    
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