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, ... , x_n) | x₁, x₂, ... , x_n∈R }.

We define the standard distance function on Rⁿ:

   If P,Q∈Rⁿ with P = (p₁,p₂, ... , p_n) and Q = (q₁,q₂, ... , q_n), then
   Dist(P,Q) = √[ (p₁ - q₁)² + (p₂ - q₂)² + ... + (p_n - q_n)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 a₀∈R with a₀ > 0.
For k ≥ 1 we define

   S_k = B(P_k,a_k) ∩ S_k-1,
   where P_k∈S_k-1 and 0 < a ≤ a_k-1.

Prove that S_k ≠ ∅ for 1 ≤ k ≤ n.


  

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