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 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 superior via the extra explanation of some of the proofs, extra explanatory comments, and clearer notation.
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Additional resources for A First Course in Real Analysis (2nd Edition) (Undergraduate Texts in Mathematics)
To find the limit of g2(X) as x -. 2l using on each occasion the one-sided version. By applying the theorems on limit of a constant, obvious limit, limit of a sum, and limit of a product, we see that lim 3(x + 2)(x 2 + 4) = 3 ,4,(4 + 4) = 96. x-2+ Similarly, lim 2(x 2 + 2x x..... 2+ + 4) = 2(4 + 4 + 4) = 24. ) Now, using limit of a quotient and the limit of composite functions, we obtain lim g2(X) = x ..... 2+ 96 f%24 = ~24 J4 = 2. Hence, limx _ 2 + f(x) = 2 and so lim x_2f(x) = 2. The function f is continuous at 2 since f(2) = 2 and f(x) -.
16) to get 3x -7 < 8, the original inequality. Since the hypothesis that x is less than 5 implies the original inequality, we have proved the result. The important thing to notice is that the proof consisted of reversing the steps of the original argument which led to the solution x < 5 in the first place. So long as each step taken is reversible, the above procedure is completely satisfactory for obtaining solutions of inequalities. By means of the symbol -, we can give a solution to this example in compact form.
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 . Since the set of natural numbers N is identical with the set of positive integers, the fact that a natural number is positive follows from the Axiom of inequality and the definition of N.
A First Course in Real Analysis (2nd Edition) (Undergraduate Texts in Mathematics) by Murray H. Protter, Charles B. Morrey Jr.