SOLUTION: Let C be a compact set in {{{R^n}}} , and let S be a subset of C such that for any x and y in S, {{{abs(abs(x-y)) >= 1}}}. Prove that S is finite.
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-> SOLUTION: Let C be a compact set in {{{R^n}}} , and let S be a subset of C such that for any x and y in S, {{{abs(abs(x-y)) >= 1}}}. Prove that S is finite.
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Question 1030129: Let C be a compact set in , and let S be a subset of C such that for any x and y in S, . Prove that S is finite. Answer by robertb(5830) (Show Source):
You can put this solution on YOUR website! Suppose C is infinite.
Then by the Bolzano-Weierstrass theorem, since C is bounded (and closed!), C will have a limit point that is contained in C.
This implies that there will be a positive integer n>1 such that for an infinite number of elements x in C.
Contradiction, because it should be that .
Hence C should be finite.
