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Unformatted text preview: Answers to Exam 1 1. Definitions on sequences: (a) { x n } → L if, given ε > 0, there exists N > 0 such that  x n L  < ε for all n > N . (b) { x n } has an accumulation point p if, for all ε > 0 and N > 0, there exists n > N such that  x n p  < ε . (c) { x n } is Cauchy if, for all ε > 0, there exists N > 0 such that  x n x m  < ε for all m , n > N . 2. Given S = { . 9 , 1 . , . 99 , 1 . 01 , . 999 , 1 . 001 ,... } . (a) sup S is the least upper bound of S , i.e. , the number B such that B ≥ s n for all n ; and if B is any other number with the same property, then B > B . Here B = 1 . 01. inf S is the greatest lower bound of S , i.e. , the number b such that b ≤ s n for all n ; and if b is any other number with the same property, then b < b . Here b = . 9. (b) { s n } → 1. Discussion: Notice that x 1 and x 2 are within 0 . 1 of 1, x 3 and x 4 are within 0 . 01 of 1, etc. Proof: Let ε > 0 be given. Choose M > 0 such that 1 / 10 M < ε , and let N = 2 M . Then  s...
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 Spring '07
 thieme
 Order theory, Multiplicative inverse, Identity element, additive inverse, multiplicative identity axiom

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