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WebNotes Theorem Rudin Theorem Proposition 2.1 Proposition 1.18 Theorem 2.2 Theorem 1.19 Proposition 2.3 Theorem 1.20(a) Archimidean Property Corollary 2.4 Theorem 1.20(b) Density of the rationals Proposition 2.5 Section 1.1 Irrationality of root 2 Proposition 2.6 Theorem 1.11 Proposition 2.7 Theorem 2.24(a) Proposition 2.8 Theorem 2.24(c) Lemma 3.1 Sandwich lemma Lemma 3.2 Lemma 3.3 Theorem 3.2(b) Lemma 3.4 Corollary 3.5 Theorem 3.3(a) Lemma 3.6 Theorem 3.2(c) Proposition 3.7 Theorem 3.3(c) Proposition 3.8 Theorem 3.3(d) Theorem 3.9 Theorem 3.3 Arithmetic properties of convergent sequences Theorem 3.10 Theorem 1.21 Theorem 3.11 Arithmetic of sequences converging to infinity Theorem 3.12 Theorem 3.14 Monotone convergence theorem Proposition 3.13 Continuity of nth roots Theorem 3.14 Theorem 2.23 A set is closed iff its complement is open Proposition 3.15 Theorem 2.24(b) The intersection of closed sets is closed Proposition 3.16 Theorem 2.24(d) Finite unions of closed sets are closed Theorem 4.1 Theorem 4.4 Arithmetic of limits of functions Theorem 4.2 Theorem 4.2 Limits of functions related to limits of sequences Proposition 4.3 Corollary to 4.2 Limits of functions are unique Proposition 4.4 Arithmetic of functions continuous at a point Theorem 4.5 Theorem 4.9 Arithmetic of continuous functions Proposition 4.6 Exponential functions are continuous Proposition 4.7 Power functions are continuous Proposition 4.8 Theorem 5.2 Differentiable functions are continuous Proposition 4.9 Theorem 5.3 Arithmetic of differentiable functions Proposition 4.10 Theorem 4.7 Composition of continuous functions is continuous Proposition 4.11 Theorem 5.5 Chain rule for differentiation Theorem 5.1 The existence of the complex field Theorem 5.2 Theorem 1.35 The Cauchy-Schwartz Inequality Lemma 5.3 Positive definite form and the discriminant Corollary 5.4 Theorem 1.37 Triangle inequality for the Euclidean norms Corollary 5.5 Example 2.16 The Euclidean spaces are metric spaces Proposition 5.6 Theorem 3.2(b) Limits are unique in metric spaces Proposition 5.7 Continuity of functions in terms of limits of sequences Proposition 5.8 Sandwich lemma for metric spaces Proposition 5.9 The projection and embedding maps in Euclidean spaces Proposition 5.10 Sums, products and quotients of real-valued functions Proposition 5.11 Composition of continuous functions Proposition 5.12 A set is open iff its complement is closed Proposition 5.13 Open balls are indeed open sets Proposition 5.14 Equivalent form of continuity using preimages of open sets Corollary 5.15 Equivalent form of continuity using preimages of closed sets Proposition 5.16 Finite sets are closed Proposition 5.17 Unions of open sets are open Proposition 5.18 Intersections of closed sets are closed Proposition 5.19 Finite unions of closed sets are closed Proposition 5.20 Finite intersections of open sets are open Theorem 5.21 The connected subsets of the reals Theorem 5.22 Continuous images of connected sets are connected Theorem 5.23 Intermediate Value Theorem Proposition 6.1 Subsequences of convergent sequences Theorem 6.2 Bolzano Weierstrauss theorem Corollary 6.3 Closed bounded sets of real numbers are sequentially compact Proposition 6.4 Sequentially compact sets are closed and bounded Theorem 6.5 Sequentially compact sets in Euclidean spaces Theorem 6.6 Continuous images of sequ. compact sets are sequ. compact Corollary 6.7 Continuous images of closed bounded sets in Euclidean spaces Corollary 6.8 Continuous functions on sequ compact sets are bounded Lemma 6.9 Closed subsets of sequ compact are sequ compact Proposition 6.10 Inverses of continuous bijections from sequ compact sets Proposition 6.11 Continuous functions have maximimums on sequ cpt sets Corollary 6.12 Continuous functions also attain their minimums Lemma 6.13 A differentiable function has zero derivative at a maximum Proposition 6.14 Rolle's Theorem Theorem 6.15 The Mean Value Theorem Corollary 6.16 A diff'ble function is increasing iff its deriv. is non-neg Corollary 6.17 A function with a strictly pos deriv is strictly increasing Proposition 6.18 The Inverse Function Theorem
Analysis WebNotes by John Lindsay Orr.
Comments to the author:
jorr@math.unl.edu
All contents copyright (C) 1995 John L. Orr
University of Nebraska--Lincoln
All rights reserved
Last modified: December 1, 1995