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Question 1 1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2....

1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2. [25 pts] Give an example of each of the following, or state

Question 1

1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2. [25 pts] Give an example of each of the following, or state one or more theorems which show that such an example is impossible: a. A countable collection of nonempty closed proper subsets of R whose union is open. b. A nonempty bounded subset of R with no cluster points. c. A convergent sequence with two convergent subsequences with distinct limits. d. A function which is continuous at a = 3 but is not differentiable at a = 3. e. A function which is differentiable at a = 3 but is not continuous at a = 3. 3. [30 pts] Prove or disprove each of these: a. The union of any collection of open sets is open. b. The intersection of a finite collection of open sets is open. c. If A is nonempty and bounded and B = {\al: a € A}, then sup B = |sup Al. 4. [10 pts] Use mathematical induction to show that for every n e N, for any set SCR such that |SI = n, S is bounded. (i.e., show that every finite set is bounded.) 5. (20 pts] Let A, B be non-empty bounded subsets of R. Let C = {[ab] : a € A, B E B}. State a theorem that expresses (in separate statements) infC and sup C in terms of inf A, inf B, sup A, and sup B. Prove one of those statements. 6. (20 pts] Use the definition of limit of a sequence to show that lim 3n2+4 3. n- na+5n 7. (20 pts] Prove that if {an} is a monotone decreasing bounded sequence, then {an} converges. 8. (20 pts] Use the definition of limit of a function to prove that f(x) is continuous at x = 1. 9.C-10 9. [10 pts] Prove that if f, g are differentiable at a, then f .g is differentiable at a and [f.g)'(a) = f'(a)g(a) + f(a)g(a). 10. [20 pts] Suppose that f, g are continuous functions from 1-1, 1) to R with -1 <f(x) <1Vx € (-1,1], and g(-1) = 1 and g(1) = -1. Prove that there exists a point c € (-1, 1) such that f(0) = g(c). 11. (20 pts] Prove that f(x) = x2 is Darboux integrable on [3, 5] by showing U(f,P) - L(f,P) < e for some uniform partition P.
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