hw5 sol_2a_5

hw5 sol_2a_5 - Tufts University Department of Mathematics...

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Tufts University Department of Mathematics Math 136 Homework 5 problems 2(a) and 5. 2008. Notes: I’m putting some review of inFma and suprema here. Some of the things I’m saying you will already know, but they bear repetition, because they are important, and you are losing more points than you need to if you don’t keep them in mind. Given a nonempty set A of real numbers, if there is a number x A that is greater than all other numbers in A , we say x is the maximum of the numbers in A , or x = max A . There may not be a maximum for A . If A is bounded from above—that is to say, there exists M such that y < M for all y A —then it is a property of the real numbers that there is a supremum of A : a number x R such that y < x for all y A and y is less than or equal to any other upper bound of A (that is to say ( y A ( y z )) = x z ). The supremum x = sup A of A may not be in A . That is to say, it may not be a maximum. This is the whole point of suprema (and inFma), that they might not be attained within the set in consideration. If we had no need for this additional generality, then we would just stick with maxima and minima. So remember, suprema and inFma are generally not attained. It is only in very special situations that they are. It is so special when a supremum or inFmum is attained (making it a maximum or a minimum), that if you believe a supremum or inFmum is attained, you must explain why. (In this case, usually the set is the
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hw5 sol_2a_5 - Tufts University Department of Mathematics...

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