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# hwk1 - ity b Use this deﬁnition to show that lim n →∞...

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MATH2400: Mathematical Analysis First Semester 2010 05/03/2010 Assignment Number 1 Problem 1 (4 points) Prove that if m and n are natural numbers and m 2 n 2 < 2, then ( m + 2 n ) 2 ( m + n ) 2 > 2 . Problem 2 (4 points) Prove (i.e. use an ε , N - argument): lim n →∞ n + 4 n = 1. Problem 3 (4 points) The elements of a logistic sequence { a n } are defined implicitly (for n N ) as follows: a n +1 = ka n (1 - a n ) , where k [0 , 4] and a 0 [0 , 1] are given. a) Prove that if k [0 , 1) and a 0 [0 , 1], then a n 0. b) Write a few lines about the occurrence of the logistic sequence in biology. Your lines should not be from Wikipedia. Problem 4 (5 points) a) Give a reasonable, precise definition for what lim n →∞ a n = and lim n →∞ a n = -∞ should mean. (This is sometimes described as the sequence diverging to infinity, or minus infin-
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Unformatted text preview: ity). b) Use this deﬁnition to show that lim n →∞ n 7 + 18 n 4 = ∞ . Problem 5 (3 points) Calculate sup Ω and inf Ω for the following subsets of R . (One or both may not exist). a) Ω = [-5 , 4) ∪ { 12 } ; b) Ω = Q ∩ (-1 , 3); c) Ω = { x ∈ R : | 3 x + 7 | > 2 } . Problem 6 (2 bonus points) Explain the mathematics behind the following (it’s not that tricky, the point is to explain it in a mathematically precise way): http://www.milaadesign.com/wizardy.html Due: 3PM, Wednesday, 17/3/2010. Current assignments will be available at http://www.maths.uq.edu.au/courses/MATH2400/Tutorials.html...
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