hw3soln - MATH 410 Homework 3 Solutions Section 2.2 1 For...

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Unformatted text preview: MATH 410: Homework 3 Solutions Section 2.2 1. For each of the following statements determine whether it is true or false and justify informally: (a) Every bounded sequence converges. Solution: False. For example { (- 1) n } . (b) A convergent sequence of positive numbers has a positive limit. Solution: False. For example braceleftbig 1 n bracerightbig . (c) The sequence { n 2 + 1 } converges. Solution: False. It is unbounded. (d) A convergent sequence of rational numbers has a rational limit. Solution: False. Since Q is sequentially dense in R we know √ 2 is the limit of a sequence of rationals. (e) The limit of a convergent sequence in the interval ( a,b ) also belongs in ( a,b ). Solution: False. For example braceleftbig a + b- a 2 n bracerightbig . 2. Show that every x ∈ R is the limit of a sequence of irrational numbers. Solution: The irrationals are dense in R and dense implies sequentially dense. Section 2.3 1. Justify whether each of the following sequences is monotone....
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This note was uploaded on 02/27/2012 for the course MATH 2 taught by Professor Staff during the Spring '08 term at Maryland.

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hw3soln - MATH 410 Homework 3 Solutions Section 2.2 1 For...

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