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05sprg-test2

# 05sprg-test2 - MAT 371 March 9 2005 Advanced Calculus Test...

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MAT 371 Advanced Calculus /120 March 9, 2005 Test 2 name 1 2 3 4 5 6 20 20 20 20 20 20 1. a. Define limit point (of a set) and limit (of a function). b. Define open set and closed set . c. State the ε - δ -definitions of continuous and uniformly continuous 2. a. Without proof, find all limit points for each of the following sets (as subsets of R ). (i) Z . (ii) Q . (ii) [0 , 1). (iv) { ( - 1) n (1+ n ) n : n Z + } . Bonus : { sin n : n Z + } . b. Working from the definition, prove that every constant sequence converges. c. Prove that the sequence ( a n ) n =1 defined by a n = ( - 1) n n +1 n does not converge. 3. Suppose that M R and ( a n ) n =1 is a sequence in R that converges to L R . Prove that if for all n , a n < M , then L M . 4. a. Suppose F X is a closed set in a metric space X . Prove that the complement O = { x X : x F } of F is an open set. b. Suppose that { F α : α Λ } is a collection of closed subsets of a metric space X . Prove that the intersection α Λ F α is a closed subset of X .
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