Download e-book for iPad: Basic Real and Abstract Analysis by John F. Randolph

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By John F. Randolph

ISBN-10: 1483231380

ISBN-13: 9781483231389

This 1968 version covers the next subject matters: Orientation; units and areas; sequences and sequence; degree and integration; degree concept; continuity; derivatives; and stieltjes integrals.

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Example text

If a and b are real or complex numbers, then , \b\ \a + b\ ^ \a\ \ + \a + b\ 1 + M 1 -r- |Z>[ * [Hint: Contrapositive proof or use elementary calculus to show that φ is an increas­ ing function and (with \b\ > 0 ) / i s a decreasing function where 9(0=1- 1 1+ t and / W = ç

With a < b given real numbers (or points), (a, b) = {x | a < x < b} is an open interval and [a, i>] = {x | a ^ x ^ b} is a closed interval (even if a = b). Closed-open and open-closed intervals are O, b) = {x | a ^ x < b} (a, b] = {x\a <χ*ζ a ( b ) E Ì E ) ( ì and b}. All of these intervals have length b — a. Whether (a, b) represents an open interval, or (later on) the plane rectangular coordinates of a point, or an ordered pair of numbers, or a complex number, or a vector must be judged by the context in which it appears.

However, Fig. 3 alone is not considered a proof (but certainly suggests a proof) of the t This form will not be used, however, until the sine and cosine functions are defined in Section 7-3. 42 Ch. 1 Orientation Theorem 2. Ifzi and z2 are complex numbers and r^ > 0 are such that z2e{z\\z-zl\ (13) 0 such that (14) {z \\z - z2\ < r2} a {z \\z - z j < r j . Proof. From (13) it follows that \z2 — zx\ 0. Then from (9) \z — z2\ \z — z j — \z2 — zt\ < r2 => \z - zx\ < \z2 - z j + (rx - \z2 - zx\) = rx which in turn implies that (14) holds for this number r2 .

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Basic Real and Abstract Analysis by John F. Randolph

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