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

Many alterations were made during this moment version of **A ** **First direction in genuine Analysis.** the main obvious is the addition of many difficulties and the inclusion of solutions to many of the odd-numbered workouts. The book's clarity has additionally been better by way of the additional explanation of a number of the proofs, extra explanatory comments, and clearer notation.

**Read Online or Download A First Course in Real Analysis (2nd Edition) (Undergraduate Texts in Mathematics) PDF**

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**Extra info for A First Course in Real Analysis (2nd Edition) (Undergraduate Texts in Mathematics)**

**Sample text**

For limits at infinity, a theorem on obvious limits takes the following form. 14 (Obvious limit). If f(x) = l/x for all x # 0, then lim f(x) = 0, x-s co lim f(x) = 0, x-e +00 lim f(x) = O. 7 on composite functions. 15 (Limit of a composite function). R 1. Iff is continuous at Land g(x) -+ L as x -+ +00, then lim f[g(x)] = f(L) . x-+oo Remarks. 15 may also be stated with x -+ -00 or x -+ 00 . 2 has three analogs according as x -+ 00, x -+ +00, or x -+ -00 . 7, respectively. 5 and the function f : x -+ xj(x + 1), x # -1, we see that f increases without bound as x tends to - 1from the left.

Any nonempty set T of natural numbers contains a smallest element. PROOF. Let k be a member of T. We define a set S of natural numbers by the relation S = {p: pET and p ~ k}. The set S contains a portion (not necessarily all) of the set consisting of the natural numbers {1, 2, 3, ... , k - 1, k}. Thus since S is finite it has a smallest element, which we denote by s. We now show that s is the smallest element of T. First, since s E Sand SeT, then SET. Suppose t is any element of T different from s.

F(x) = 3x - 2 { 2x - 1, 8. f(x) = {C ~ 2 1 < x < 2, 2 ~ x < 4, 2 ;:: : 2r 1 < x < 2, , x 2 /2, 9. f(x) = { to. f(x) = { ( : : : x:f. 2, :r . ), x ~3 2 < x < 3, x:f. 0, X 12. f(x) = 3" x < 4, a sm - , x { a=2 x=2, x - 9, 11. f(x) = a= 2 2 ~ x < 5, x: - 8, x - 4 4, a=2 x:f. 0, a=O a=O x=O, 13. Given f(x) = tan x, -rc/2 < x < rc/2, g(x) = (rc/2) - x. What conclusion can we draw about lim f[g(x)] when a = rc/4? Same problem when a = rc/2, when a = O. 14. 4 on the limit of equal functions for limits from the left.