Science:Math Exam Resources/Courses/MATH220/April 2005/Question 11
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Question 11 

Let M be the set Find, with proof, inf(M). 
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Hint 

Recall that inf(M) denotes the infimum of the set M, that is the largest real number that is smaller than any element of the set M. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. The condition is equivalent to the conditions This means, we can rewrite the set M as being The infimum of a set being the largest real number that is smaller or equal to all the numbers of the set, it seems to be a good idea to think that √5 is a good candidate for the infimum of the set M. Since √5 is clearly smaller than all the numbers of the set M, we only need to show that it is the largest real number with that property. Assume that there is a number m with that property as well, that is strictly larger than √5, that is We'll simply show that such a m cannot exist and hence √5 is the infimum of the set M. Indeed, if m is strictly larger than √5 than it means that the interval is not empty. Since nonempty interval of real numbers contain infinitely many numbers, we can find two distinct numbers in that interval and , that is Now a property of the rational numbers is that you can always find a rational number between any two distinct real numbers. That means there must exist a rational number q between and , so we have but in particular that means that q is in the interval (√5,√5) and since it is rational, q is an element of the set M. But by construction, q < m which contradicts the assumption that m could be a lower bound of the set M. This means that such a number m cannot exist and hence the infimum of the set M has to be √5. 