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)**

**Sample text**

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.