MTH 320 Midterm: Math 320 Exam 1
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Summer 2015: (5 + 10 points) let a be a nonempty set of real numbers. (a) state the de nition of sup a. Then = sup a provided: (i*) > a for all a a. That is, is an upper bound of a. (ii*) if is any upper bound of a, then . Show that = sup a. (note: in class we used this alternate characterization to prove several important results. ) Since (i) and (i*) are the same we need only show (ii) implies (ii*). So let be any upper bound of a. If is less than , then we let = > 0. Summer 2015: (15 points) suppose that a and b are bounded nonempty subsets of real numbers with sup a < sup b. Show that there is an element b b that is an upper bound for a.
