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

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

Many alterations were made during this moment version of A First direction in actual Analysis. the main visible is the addition of many difficulties and the inclusion of solutions to many of the odd-numbered routines. The book's clarity has additionally been more desirable by way of the additional explanation of the various 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)

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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. If f(x) ~ g(x) ~ h(x) on this interval, and if Iimf(x) then lim x _ a g(x) = L, lim h(x) = L, = L. PROOF. 2. Limits 41 and whenever 0 < Ix - al < 82 , Ih(x) - LI < e In other words, L - s

Limits 41 and whenever 0 < Ix - al < 82 , Ih(x) - LI < e In other words, L - s

1, and y = ! 1. Since the distance from x = 1 to x = t is smaller than the distance from x = 1 to x = t, we select ~ = 1 - t = t. We make the important general observation that when a value of ~ is obtained for a given quantity e, then any smaller (positive) value for ~ may also be used for the same number e. 0 Remarks. For purposes of illustration, we assume in this chapter that::r;. is defined for all x ~ 0 if n is even and that ::r;. is defined for all x if n is odd. , with n a natural number, do not follow from the results given thus far.

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