practicetest

practicetest - (c A sequence with subsequences converging...

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MAA 4224–Introduction to Analysis, Fall 2011 Practice Test These are practice problems for the midterm. The midterm itself will consist of 5 problems, from which you choose 4 to be graded. 1. Negate the following statements: (a) For any a A and b B , we have a < b . (b) Given ± > 0, there exists an N N such that | a n - a | < ± whenever n N . 2. If A,B R and sup A < sup B which of the following is true? (a) A B . (b) There is a b B such that b > sup A . 3. Use the Archimedean property to show that \ n =1 [ n, ) = . 4. Which of the following sets is countable? Justify your answers. (a) The odd integers. (b) The rational numbers strictly between 0 and 1. (c) The irrational numbers strictly between 0 and 1. 5. Which of the following sequences converges? Give complete proofs using statements involving ± and N . (a) (3 , 3 . 1 , 3 , 3 . 01 , 3 , 3 . 001 ,... ). (b) a n = n n +5 . (c) (1 , 2 , 3 ,... ). (d) a n = 1 + 1 / 2 + 1 / 4 + ... + 1 /n 2 . 1
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6. Give an example, or prove that none can be found. (a) A convergent sequence that is not monotone. (b) A Cauchy sequence that has an unbounded subsequence.
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Unformatted text preview: (c) A sequence with subsequences converging to each of the integers. (d) A sequence ( a n ) so that ( a n ) converges, but ∞ X n =1 a n diverges. 7. Prove or give a counter-example. (a) If ( a n ) converges to zero, then for any b , ( a n + b ) converges to b . (b) If ( a n ) is monotone increasing, then it is bounded. (c) If b k is a sequence of positive integers, then a n = n X k =1 b k is a sequence monotone increasing. 8. State the following theorems and properties and their negations. (a) Monotone Convergence Theorem. (b) Archimedean property. (c) Nested Interval Property. (d) Cauchy Theorem. 9. State the following definitions and their negations. Give an example, and a non-example for each one. (a) A Cauchy sequence. (b) The supremum of a set. (c) A nested interval. (d) A convergent series. 2...
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practicetest - (c A sequence with subsequences converging...

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