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 genuine Analysis. the main visible 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 more suitable through the additional explanation of the various proofs, extra explanatory feedback, and clearer notation.

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Additional resources for A First Course in Real Analysis (2nd Edition) (Undergraduate Texts in Mathematics)

Sample text

Continuity and Limits 9. 5. 10. 6. *11. Let f(x) = x sin(l lx) for -n ~ x ~ n, x ~ O. ) (a) Prove that Iim,,_of(x) = O. (b) Define g(x) == f(x). By observing that g(x) = 0 for x, = ± linn, n = I, 2, . , conclude that f[g(x)] is not defined for x = x e- Note that x, -+ 0 as n -+ 00. (c) Using the result of Part (b) show that lim h(x) ,,-0 does not exist where h(x) = f[g(x)]. 7. 7 to show 0 sin ~ x -n ~ x ~ n, x ~ 0 } x=o that lim,,_o H(x) = 0 where H(x) = F[g(x)]. 12. R 1 . R 1• Show that F is continuous.

4). Of course, infinity is not a number, but we use the symbol (7, (0) to represent all numbers larger than 7. We also use the double inequality 7

K because the assumption k E S implies these b,are unique. Now, and so (k + 1) E S. The set S is inductive and therefore S = N. The theorem is true for every natural number n. 20. For each finite sequencea l ' a2' ... , an there is a unique sequence c l , c2, ... , Cn such that and for i = 1, 2, .. , n - 1 if n > 1. 19 and is left to the reader. 20. Its proof, which employs the Principle of mathematical induction, is left to the reader. 1. 21. If a1' a2' ... , am' am+1, ... , am+n is any finite sequence, then m+n m m+n i=l i=l i=m+1 L a, = L a, + L and n a, = n ai ' n m+n i=l ( m i=l ) (m+n i=m+1 a, ) ai .