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Let

S

be an infinite set of integers containing zero, and such that the distances between successive number never exceed a given fixed number. Consider the following procedure: Given a set

X

of integers we construct a new set consisting of all numbers

x \pm s,

where

x

belongs to

X

and s belongs to

S.

Starting from

S_0 = \{0\}

we successively construct sets

S_1, S_2, S_3, \ldots

using this procedure. Show that after a finite number of steps we do not obtain any new sets, i.e.

S_k = S_{k_0}

for

k \geq k_0.

75

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