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Unformatted text preview: Kimberly Zhang Analysis Useful Things a + b ≤ a + b  a – b  ≤ a – b Definitions L is an upper bound for S iff L ≥ s for all s S. ∈ L is the least upper bound for S iff L is an upper bound for S and if M is also an upper bound for S, then L ≤ M. A sequence {a n } converges to a real number A iff for each ε > 0, there is a positive integer N such that for all n ≥ N, we have a n – A < ε. A sequence {a n } is Cauchy iff for any ε > 0, there is a positive integer N such that for all m,n ≥ N, we have a m – a n  < ε. Let S be a set of real numbers. A real number A is an accumulation point of S iff every neighborhood of A contains infinitely many points of S. Axiom of Completeness: Any nonempty subset of R that is bounded above has a least upper bound. Monotone Convergence Theorem: A monotone sequence is convergent iff it is bounded. Let ƒ: D R with x an accumulation point of D. Then f has a limit at x iff for each ε > 0 there is a δ > 0 such that if 0 < x – x...
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 Spring '08
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 Calculus, Continuous function, accumulation point, Kimberly Zhang

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