document.write( "Question 1030129: Let C be a compact set in $\"R%5En\"$ , and let S be a subset of C such that for any x and y in S, $\"abs%28abs%28x-y%29%29+%3E=+1\"$ . Prove that S is finite. \n" ); document.write( "

Algebra.Com's Answer #645105 by robertb(5830) $\"\"$ $\"About$

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Suppose C is infinite.\r
\n" ); document.write( "\n" ); document.write( "Then by the Bolzano-Weierstrass theorem, since C is bounded (and closed!), C will have a limit point $\"x%5B0%5D\"$ that is contained in C.\r
\n" ); document.write( "\n" ); document.write( "This implies that there will be a positive integer n>1 such that $\"abs%28abs%28x-x%5B0%5D%29%29+%3C+1%2Fn+%3C+1\"$ for an infinite number of elements x in C.\r
\n" ); document.write( "\n" ); document.write( "Contradiction, because it should be that $\"abs%28abs%28x-x%5B0%5D%29%29+%3E=+1\"$ .\r
\n" ); document.write( "\n" ); document.write( "Hence C should be finite.
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