By David V. Widder

Vintage textual content leads from simple calculus into extra theoretic difficulties. targeted strategy with definitions, theorems, proofs, examples and routines. themes contain partial differentiation, vectors, differential geometry, Stieltjes imperative, endless sequence, gamma functionality, Fourier sequence, Laplace remodel, even more. a variety of graded workouts with chosen solutions. 1961 edition.

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**Sample text**

Let A be any countable set and lei B cA. If B is finite, there is nothing to prove. Thus suppose that B is infinite. Then A is countably infinite. Let (an) be an enumeration of A. IY. 18) Theorem. The Cartesian product NxN is a countable set. Proof. , NxN. One way to do this is to define the mapping I from N x N onto N by I(m, n) = 2m-l(2n - 1). § 4. Cardinal numbers and ordinal numbers 23 Since each positive integer is a power of 2 [possibly the OtTl power] times an odd integer, I is onto N. We see that I is one-to-one, for otherwise there would be an integer which is both even and odd.

0 We next consider the cardinal number c;: = R. The reader is invited to look ahead to § 5 for a detailed construction of R and for the relevant properties of R that we use here. § 4. 26) Theorem. Let ]0, l[={xER:O

40) Definition. Let A and B be linearly ordered sets. An order isomorphism Irom A onto B is a one-to-one function 1 from A onto B such that x ~ y in A implies 1(x) ~ I(y) in B. We write A ~ B to mean that such an order isomorphism exists. It is easy to see that the relation ~ is reflexive, symmetric, and transitive. With every linearly ordered set A we associate a symbol, called the order type 01 A, such that two linearly ordered sets A and B have the same symbol attached to them if and only 28 if A Chapter 1.