Science:Math Exam Resources/Courses/MATH220/April 2011/Question 10 (a)
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Question 10 (a) 

Let be a bounded set of real numbers and let be a nonempty subset of A. Prove that B is also bounded and that 
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Hint 

Recall the definition of supremum and infimum of sets of real numbers. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. By definition of the supremum of a set, sup(B) is the smallest real number that is larger or equal to all the numbers in the set B. Since all the numbers of the set B are also numbers of the set A, we can conclude that sup(A) is also larger or equal to all the numbers of the set B (since it is larger or equal to all numbers in the set A) but since sup(B) is the smallest real number with that property we can conclude that Now inf(A) is a number that is less or equal than any number in the set A and so since B is a subset, it is in particular less or equal to any number in the set B. Since sup(B) is larger than any number in the set B we can conclude that Note: we could actually prove the more interesting and stronger statement: 