Science:Math Exam Resources/Courses/MATH220/April 2005/Question 11
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Let M be the set
Find, with proof, inf(M).
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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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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 non-empty 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.