By Murray H. Protter, Charles B. Morrey Jr.
Many adjustments were made during this moment version of A First direction in actual Analysis. the main obvious is the addition of many difficulties and the inclusion of solutions to lots of the odd-numbered workouts. The book's clarity has additionally been greater by way of the additional rationalization of a few of the proofs, extra explanatory comments, and clearer notation.
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Extra resources for A First Course in Real Analysis (2nd Edition) (Undergraduate Texts in Mathematics)
A(r" - + (n - l)d]/2,whereaanddareany real numbers. 1)/(r - I), where a and r are real numbers, and r L7=1 i(i + l)(i + 2) = 13. (l 14. I)d] = n[2a n(n #; 1. + 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.
Iii) 1 > O. 14 (i) a and b are both positive or both negative if and only if a · b > 0; a and b have opposite signs if and only if a· b < o. (ii) If a ;f; 0 and b ;f; 0, then a . b and alb have the same sign. 3. 15 (i) If a > band c > 0, then a C b c a c b c > be and -> -. ac < be and - <-. 16. If a > band b > c, then a > c. 15yield the following statements on inequalities, ones which are familiar to most readers. (a) The direction of an inequality is unchanged if the same number is added to or subtracted from both sides.
Remarks. 9 it is not necessary that f and 9 be defined at x = a. If f and 9 are continuous at a, we conclude from the theorem that f(a) ~ g(a). If, in the hypothesis, we assume that f(x) < g(x) in an interval about a, (excluding a itself) it is not possible to conclude that L < M . For example, letting f(x) = x 3 and g(x) = x 2 , we observe that f(x) < g(x) for -1 < x < 1, x =I O. However, f(O) = g(O) = O. 10 (Sandwiching theorem). Suppose that f, g, and h are functions defined on the interval 0 < Ix - al < k for some positive number k.