MTH 320 Midterm: Math 320 Exam 2
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Throughout this exam you may assume that a r is never the empty set: (10 points) let {an} and {bn} be bounded sequences of positive real numbers. Ifp show thatp n=1 an bn converges. n=1 an converges. Since {bn} is bounded there is an l > 0 such that for all n n we have 0 < bn < l. now an > 0 implies (1) 0 < anbn < lan, n n. The result now follows by combining (1) and (2) and invoking the comparison test. Then by the cauchy criterion for series, there exists n n such that for all n, m > n we have. The desired result now follows by the cauchy criterion for series. rjh. Summer 2015: (15 points) use an - argument to prove that f (x) = 5x2 + 1 is continuous at 3. We need to show that limx 2 x2 = 4.
