Download e-book for kindle: A First Course in Real Analysis (2nd Edition) (Undergraduate by Murray H. Protter, Charles B. Morrey Jr.

By Murray H. Protter, Charles B. Morrey Jr.

ISBN-10: 0387974377

ISBN-13: 9780387974378

Many alterations were made during this moment variation of A First path in actual Analysis. the main seen is the addition of many difficulties and the inclusion of solutions to lots of the odd-numbered routines. The book's clarity has additionally been superior by means of the additional rationalization of a number of the proofs, extra explanatory comments, and clearer notation.

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I)(n + 2)(n + 3)/4. + a)" ~ 1 + na for a ~ 0 and n a natural number . L7=1 321-1 = 3(9" - 1)/8. 15. Prove by induction that (Xl + X2 + ... + Xk)2 k = L xl + 2(XIX2 + XIX3 + ... + XIXk + X2 X3 1=1 16. 20. 17. 21. 18. 2. 19. 2. 5 We denote by N x N the set of all ordered pairs of natural numbers (m, n). State and prove a Principle of mathematical induction for sets contained in N x N. 5 An asterisk is used to indicate difficult problems. 1. Continuity Most of the functions we study in elementary calculus are described by simple formulas .

For this purpose we need the concept of a one-sided limit. fi Definition. R i . The function f tends to L as x tends to a from tbe rigbt if and only if (i) there is an open interval I in D which has a as its left endpoint and (ii) for each B > 0 there is a ~ > 0 such that I/(x) - LI < B whenever 0 < x - a < ~. 3. One-Sided Limits 43 If f tends to L as x tends to a from the right, we write f(x) -+ L x as -+ a+ and we denote the number L by lim f(x) x-+a+ or by Iim",_a +f(x). A similar definition is employed for limits from the left.

For all x E R 1 define _ {I ifxis a rational number, f (x) - 0 if x isan irrational number. Show that f is not continuous at every value of x. 22. Supposethatfisdefinedinanintervalaboutthenumberaandlimh~o(f(a + h)f(a - h)] = O. Show that f may not be continuous at a. Is it always true that Iimh~of(a + h) exists? 2. Limits The basic theorems of calculus depend for their proofs on certain standard theorems on limits. These theorems are usually stated without proof in a first course in calculus. In this section we fill the gap by providing proofs of the customary theorems on limits.

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A First Course in Real Analysis (2nd Edition) (Undergraduate Texts in Mathematics) by Murray H. Protter, Charles B. Morrey Jr.

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