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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.

  Submitted by Bractals    
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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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