MATH+301+First+Exam+2005

# MATH+301+First+Exam+2005 - Math 301 First Exam Monday, Oct....

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Math 301 First Exam Monday, Oct. 10, 2005 Try to give as much detail as possible in your answers. If you use a theorem, state clearly which theorem you are using and show that the hypotheses of that theorem apply. 1. a) State the definition of a Cauchy sequence of real numbers. b) Suppose 0 < λ < 1, and suppose m,n with m < n. Write (but do not prove!) a formula (a ratio of two terms) for λ m + λ m+1 + ………. . + λ n . c) With λ as above and C > 0, prove that C λ n → 0 as n → +∞ . (For ε > 0, find N >0 such that ……. by using logs to define N, where log is, of course,base e. ) d) Suppose {a n } is a sequence of complex numbers and |a n+1 - a n | λ n | a 1 - a 0 | , where 0 < λ < 1. Prove that the sequence is a Cauchy sequence. (Hint: use parts b,c.) 2. a) State the definition of lim inf for a sequence {a n }. b) State the Bolzano - Weierstrass Theorem. c) Prove the BW Theorem when all the points of the sequence are in [0,1]. 3. a) Prove that x is countable. (You may assume the following

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## This note was uploaded on 10/07/2009 for the course MATH 301 at Yale.

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MATH+301+First+Exam+2005 - Math 301 First Exam Monday, Oct....

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