04fall-test2

# 04fall-test2 - MAT 371 Advanced Calculus Test 2 name 1...

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MAT 371 Advanced Calculus /100 October 27, 2004 Test 2 name 1 Bonus 2 3 4 5 6 15 10 20 20 20 15 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 1. a. State the deﬁnition of Cauchy sequence . b. Brieﬂy summarize the diﬀerence between limit and limit point . c. State the deﬁnition of compact . Bonus. Give several diﬀerent arguments that show that Q [0 , 10] is not compact. 2. For each of the following subsets of R ﬁnd ( without proof ) the set of all limit points. a. (0 , 1]. b. Q . c. { ( - 1) n n n +1 : n Z + } . d. { 2 - n : n Z } . 3.a. Suppose that K, a R , and f : R 7→ R is such that for all x 6 = a , f ( x ) > K . Assuming that lim x a f ( x ) exists use the ε - δ -deﬁnition of the limit to show that lim x a f ( x ) K . b. Give an example that shows that it need not be true that lim x a f ( x ) > K . 4. a. Prove that every Cauchy sequence (in a metric space) is bounded.
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