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Sequence of subsets (Posted on 2014-09-06) Difficulty: 4 of 5

  
Definitions and Nomenclature

n is an integer > 1.

∅ denotes the empty set.

R is the set of real numbers ( the complete ordered field ).

Rn = { (x1, x2, ... , xn) | x1, x2, ... , xnR }.

We define the standard distance function on Rn:

   If P,Q∈Rn with P = (p1,p2, ... , pn) and Q = (q1,q2, ... , qn), then
   Dist(P,Q) = √[ (p1 - q1)2 + (p2 - q2)2 + ... + (pn - qn)2 ].

If P∈Rn and a∈R with a > 0, we define

   B(P,a) = { Q∈Rn | Dist(P,Q) = a }.

Problem

Given S0 = Rn and a0R with a0 > 0.
For k ≥ 1 we define

   Sk = B(Pk,ak) ∩ Sk-1,
   where Pk∈Sk-1 and 0 < a ≤ ak-1.

Prove that Sk ≠ ∅ for 1 ≤ k ≤ n.


  

No Solution Yet Submitted by Bractals    
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