1549901369-Elements_of_Real_Analysis__Denlinger_

9.4 *Two Results of Weierstrass 571 . ~ cos(l0k7rx) Proof. Define the function f(x) = L.., fk(x), where fk(x) = 2 k. k=l Claim # ...

572 Chapter 9 • Sequences and Series of Functions Apply the triangle inequality to get 1 lf(an) - f(x)I + IJ(x) - f(bn)I ;:::: l ...

9.4 *Two R esults of Weierstrass 573 after a convenient number of terms, and thus were really using polynomials to approximate f ...

574 Chapter 9 • Sequences and Series of Functions Definition 9.4.8 A continuous function f : [a, b] ---+ IR is said to be piece- ...

9.4 *Two Results of Weierstrass 575 Proof. Suppose f : [a, b] --t JR is continuous and c > 0. Since f is uniformly continuous ...

576 Chapter 9 • Sequences and Series of Functions n (c) 'tit E [xi-1, xi], f(t) = f(a) + 2.: (mi - mi-1)(t - Xi-1)+. i=l Since t ...

9.4 *Two Results of Weierstrass 577 Show that Pn(x) is a polynomial in x^2 of degree 2n-^1. Use mathematical induction to prove ...

578 Chapter 9 • Sequences and Series of Functions Corollary 9.4.20 Given any continuous f : [a, b] ---+ JR, there is a series 00 ...

9.5 *A Glimpse Beyond the Horizon 579 such as C[a, b]. But there are other norms commonly in use on C[a, b] and its subspaces; f ...

580 Chapter 9 • Sequences and Series of Functions 9.3.6 tell us that a sequence {f n} in C [a , b] converges to some f E C[a, b] ...

9.5 *A Glimpse Beyond the Horizon 581 Theorem 9.5.3 Every complete normed vector space is of second category. The reason for inc ...

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Appendix A Logic and Proofs A.1 The Logic of Propositions The theory of deductive logic, as it applies to mathematics, divides n ...

584 Appendix A • Logic and Proofs sentences may express the same proposition. For example, "John is five feet eleven inches tall ...

A. l The Logic of Propositions 585 Examples A.1.3 Some conjunctions: (a) I ate a snack and then I went to bed. (b) "I like chees ...

586 Appendix A • Logic and Proofs Definition A.1.6 The implication "If P, then Q" (or "P implies Q" ) is symbolized P =? Q and i ...

A.l The Logic of Propositions 587 ALTERNATIVE WAYS OF SAYING P::::} Q The English language provides a variety of expressions, al ...

588 Appendix A • Logic and Proofs Thus, P <=> Q is true only when P and Q have the same truth value. Sometimes we say, "Pi ...

A .l The Logic of Propositions 589 Thus, '""P has the opposite truth-value from P. Examples A.1.13 Some negations: (a) The negat ...

590 Appendix A • Logic and Proofs A necessary condition for an integer to be divisible by 12 is that it be divisible by both 4 ...