Algebra.Com's Answer #852107 by ikleyn(52779)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "If A, B are two non-empty sets of R and bounded, prove that
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document.write( "inf (A+B) = inf A + inf B
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document.write( " The proof consists of two parts.\r
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document.write( " First part of the proof\r\n" );
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document.write( "Let a = inf(A), b = inf(B).\r\n" );
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document.write( "Since the subsets A and B are bounded in R, the values 'a' and 'b' do exist and are defined properly.\r\n" );
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document.write( "The fact that a = inf(A) means that there is an infinite sequence  of elements in A\r\n" );
document.write( "which converges to 'a'.\r\n" );
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document.write( "The fact that b = inf(B) means that there is an infinite sequence  of elements in B\r\n" );
document.write( "which converges to 'b'.\r\n" );
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document.write( "Then the sequence  converges to value a+b.\r\n" );
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document.write( "This simple elementary statement is easy to prove.\r\n" );
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document.write( "It implies that \r\n" );
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document.write( " inf(A+B) <= a+b. (1)\r\n" );
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document.write( " Second part of the proof\r\n" );
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document.write( "Again, let a = inf(A), b = inf(B).\r\n" );
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document.write( "Since the subsets A and B are bounded in R, the values 'a' and 'b' do exist and are defined properly.\r\n" );
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document.write( "Since the subsets A and B are bounded in R, the set of all real numbers of the form {x+y), \r\n" );
document.write( "where x is from A and y is from B, is bounded, too.\r\n" );
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document.write( "Hence, the set of all sums (x+y) has the infinum. Let z = inf(A+B).\r\n" );
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document.write( "The fact that z = inf(A+B) means that there is an infinite sequence \r\n" );
document.write( "with elements in A and in B, which converges to z.\r\n" );
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document.write( "Notice that all are not less than 'a', and all are not less than 'b', \r\n" );
document.write( "due to the definitions of 'a' and 'b'.\r\n" );
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document.write( "It means that\r\n" );
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document.write( " z = inf(A+B) = ( lim  as i -->  ) >= a + b = inf(A) + inf(B). (2).\r\n" );
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document.write( "Inequalities (1) and (2), taken together, prove that\r\n" );
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document.write( " inf(A+B) = inf(A) + inf(B).\r\n" );
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document.write( "QED.\r\n" );
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document.write( "Solved, proved and completed.\r
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