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# final - Mathematics 131A Fall 2007 FINAL Your Name...

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Mathematics 131A Fall 2007 FINAL Your Name: Signature: INSTRUCTIONS: This is a closed-book test. Do all work on the sheets provided. If you need more space for your solution, use the back of the sheets and leave a pointer for the grader. Good luck! Point Points Problem Value Received 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 Total 90

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Problem 1. (10 points) Print your name: Let { x n } be a sequence of real numbers such that x n + as n → ∞ . Show that inf { x n ; n N } = x n 0 , for some n 0 N .
Problem 2. (10 points) Print your name: Let I R be an open interval and let f : I R be a function such that f ( x ) 0, x I . Assume that f is continuous at x 0 I . Give an ε δ proof of the fact that f is continuous at x 0 .

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Problem 3. (10 points) Print your name: Let { a n } and { b n } be two sequences of nonnegative real numbers, a n 0, b n 0. Assume that the limit lim n →∞ a n b n = L exists and that L > 0. Show that either the series n =1 a n and n =1 b n both converge or both diverge.
Problem 4. (10 points)

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final - Mathematics 131A Fall 2007 FINAL Your Name...

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