This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: David M. Bressoud Macalester College, St. Paul, MN Project NExTWI, October 5, 2006 “The task of the educator is to make the child’s spirit pass again where its forefathers have gone, moving rapidly through certain stages but suppressing none of them. In this regard, the history of science must be our guide.” Henri Poincaré “The task of the educator is to make the child’s spirit pass again where its forefathers have gone, moving rapidly through certain stages but suppressing none of them. In this regard, the history of science must be our guide.” Henri Poincaré 1. Cauchy and uniform convergence 2. The Fundamental Theorem of Calculus 3. The Heine–Borel Theorem 1. Cauchy and uniform convergence 2. The Fundamental Theorem of Calculus 3. The Heine–Borel Theorem A Radical Approach to Real Analysis , 2nd edition due January, 2007 A Radical Approach to Lebesgue’s Theory of Integration, due December, 2007 “What Weierstrass — Cantor — did was very good. That's the way it had to be done. But whether this corresponds to what is in the depths of our consciousness is a very different question … “What Weierstrass — Cantor — did was very good. That's the way it had to be done. But whether this corresponds to what is in the depths of our consciousness is a very different question … Nikolai Luzin … I cannot but see a stark contradiction between the intuitively clear fundamental formulas of the integral calculus and the incomparably artificial and complex work of the ‘justification’ and their ‘proofs’. … I cannot but see a stark contradiction between the intuitively clear fundamental formulas of the integral calculus and the incomparably artificial and complex work of the ‘justification’ and their ‘proofs’. Nikolai Luzin Cauchy, Cours d’analyse , 1821 “…explanations drawn from algebraic technique … cannot be considered, in my opinion, except as heuristics that will sometimes suggest the truth, but which accord little with the accuracy that is so praised in the mathematical sciences.” Niels Abel (1826): “Cauchy is crazy, and there is no way of getting along with him, even though right now he is the only one who knows how mathematics should be done. What he is doing is excellent, but very confusing.” Cauchy, Cours d’analyse, 1821, p. 120 Theorem 1. When the terms of a series are functions of a single variable x and are continuous with respect to this variable in the neighborhood of a particular value where the series converges, the sum S ( x ) of the series is also, in the neighborhood of this particular value, a continuous function of x . S x ( 29 = φ κ ξ ( 29 κ =1 ∞ ∑ , φ κ χοντινυουσ ⇒ Σ χοντινυουσ S n x ( 29 = φ κ ξ ( 29 κ =1 ν ∑ , Ρ ν ξ ( 29 = Σ ξ ( 29  Σ ν ξ ( 29 Χονωεργενχε ⇒ χαν μ ακε Ρ ν ξ ( 29 ασσμ αλλασϖε ϖιση βψτακινγ ν συφφιχιεντλψλαργε. Σ ν ισχοντινυουσφορν < ∞. S...
View
Full Document
 Winter '08
 JARVIS
 Math, Émile Borel

Click to edit the document details