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 Spacy Colors Lemma (Posted on 2014-12-06)

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.

 Submitted by Bractals No Rating Solution: (Hide) ```Proof that tk∈(0,1) for all k>1: t2 = a2/(2*r1) = a2/(2*a1) ≤ a1/(2*a1) = 1/2 < sqrt[ 1/2 ] tk < sqrt[ 1/2 ] ⇒ tk*tk < 1/2 ⇒ 1/2 < 1 - tk*tk ⇒ sqrt[ 1/2 ] < sqrt[ 1 - tk*tk ] ⇒ ak*sqrt[ 1/2 ] < ak*sqrt[ 1 - tk*tk ] ⇒ ak*sqrt[ 1/2 ] < rk ⇒ ak+1*sqrt[ 1/2 ] < rk ⇒ ak+1/(2*rk) < 1/(2*sqrt[ 1/2 ]) ⇒ tk+1 < sqrt[ 1/2 ] Therefore, tk < sqrt[ 1/2 ] < 1 for all k>1. Clearly, tk > 0 for all k>1 unless a tk-1 = 1. Therefore, tk∈(0,1) for all k>1. QED ```

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