Rudin 【Principle of Mathematical Analysis】Notes & Solutions
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04-25-2012, 11:18 AM
(This post was last modified: 11-26-2018 04:11 PM by elim.)
Post: #11
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1.17-18 Ordered Fields-- Rudin [Principle of Mathematical Analysis] Notes
1.17 Definition An ordered field is a field $F$ which is also ordered:
$\quad$(i) $\ \forall x,y,z\in F\quad (x < y)\implies (x+z < y+z)$ $\quad$(ii) $\forall x,y\in F\quad\quad\; (x>0)\wedge (y>0)\implies (xy > 0)$ If $x>0$, we call $x$ possitive; if $x < 0$, we call $x$ negative. $\mathbb{Q}$ is an example of ordered field. 1.18 Proposition Any ordered Field has the following properties: (a) $(x > 0) \Leftrightarrow (-x < 0)$ (b) $(x > 0)\wedge (y < z)\implies (xy < xz)$ (c) $(x < 0)\wedge (y < z)\implies (xy > xz)$ (d) $(x\ne 0)\implies (x^2 > 0)$. In particular, $1 > 0$ (e) $(0 < x < y)\implies (0 < 1/y < 1/x)$ Proof (a) $(x > 0)\implies (0 = -x + x > -x + 0 = -x)$, inversely, $\quad\;\; (x < 0)\implies (0 = -x+x < -x+0)$ (b) $(x>0)\wedge (z > y)\implies (x>0)\wedge (z-y > y-y = 0)$ $\quad\;\;$thus$\; xy + 0 < xy + x(z-y) = xz$ (c) By (a),(b) & Proposition 1.16c, $-(x(z-y)) = (-x)(z-y) > 0$ $\quad\;\;$so $x(z-y)< 0$ or $xz < xy$ (d) If $x > 0$ then Def. 1.17(ii) gives $x^2 > 0$, $\quad\;\;$if $x < 0,\;$then $-x > 0,\;\; x^2 = (-x)^2 > 0$ by (a) & 1.16(d) (e) If $((x>0)\wedge (1/x\le 0))\,$then$\; 1 = x(1/x) \le 0$ contradict to (d). $\quad\;\;$So $1/x > 0\,($likewise $1/y > 0)\,$thus$\,((1/x)(1/y) > 0) \wedge (x < y)$ $\qquad\qquad\qquad\qquad \implies 1/y = x(1/x)(1/y) < y(1/x)(1/y) = 1/x.$ |
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04-26-2012, 02:03 PM
(This post was last modified: 01-29-2019 04:41 PM by elim.)
Post: #12
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1.19-1.21 The Real Numbers-- Rudin [Principle Math Analysis] Notes
The Real Field
The core of the chapter: the existence theorem (of Real Field) 1.19 Theorem There exists an ordered field $\mathbb{R}$ which has the least-upper-bound propery. Moreover, $\mathbb{R}$ contains $\mathbb{Q}$ as a subfield. The 2nd statement means that $\mathbb{Q}\subset \mathbb{R}$ and the addition and multiplication in $\mathbb{R}$, when applied to members of $\mathbb{Q}$, coincide with the usual operations on $\mathbb{Q}$; also, the positive rational numbers are positive elements of $\mathbb{R}$ The members of $\mathbb{R}$ are called real numbers. The proof of the theorem, actually constructs $\mathbb{R}$ from $\mathbb{Q}$, will presented as an Appendix to Chap.1. We'd like to show how the LUB (least-upper-bound) property implies the following: 1.20 Theorem (a) $\forall x\in\mathbb{R}^+,\forall y\in\mathbb{R}\; \exists n\in\mathbb{N}\quad (nx > y)$ (b) $ (x,y \in\mathbb{R})\wedge (x < y) \implies (x,y)\cap \mathbb{Q} \ne \varnothing $ Proof $\quad$(a) If no such an $n$, then $A = \{nx | n\in\mathbb{N}\}$ has upper bound $y$ thus $\alpha = \sup A$ exists in $\mathbb{R}$. Since $x>0$, $\alpha - x$ is not an upper bound of A hence $\alpha -x < mx \le \alpha$ for some $m\in\mathbb{N}$, but then $\alpha < (m+1)x$ and so $\alpha$ cannot be an upper bound of $A$. $\quad$(b) Take(by(a)) $n\in\mathbb{N}$ such that $n(y-x) > 1$. Likewise we have $m_1,m_2\in\mathbb{N}$ such that $m_1>nx,\; m_2 > -nx$ and so $-m_2 < nx < m_1$. Thus $m-1\le nx < m$ for some integer $m$ with $-m_2 < m \le m_1$. Combine all these we see that $nx < m\le 1+nx < n(y-x)+nx = ny$. Since $n$ is positive, we get $x < \frac{m}{n} < y\quad\square$ Now the existence of $n$th roots of positive reals 1.21 Theorem $\forall x\in\mathbb{R}^+ \ \forall n\in\mathbb{N}^+\; \exists ! y\in\mathbb{R}^+\; (y^n = x)\quad$ $\qquad$[We write $y = \sqrt[n]{x}$ or $y = x^{1/n}$] Proof The uniqueness of $n$th root is obvious: $y_1^n < y_2^n\small\,(0< y_1 < y_2 )$ Let $t = x/(1+x)$, then $0 < t < \min(x, 1)$ thus $t^n \le t < x$ and so $E = \{t \in\mathbb{R}: t > 0, t^n < x \} \ne \varnothing.$ Clearly that $1+x$ is an upper bound of $E$ and so $y = \sup E$ exists. We need the inequality $b^n - a^n < n(b - a)b^{n - 1}\quad (0 < a < b)$. If $y^n < x$, choose $h\in (0,1)$ such that $h < \frac{x - y^n}{n(y+1)^{n -1}}$, then for $a = y, b= y +h$ the inequality becomes $\quad\quad(y +h)^n - y^n < nh(y +h)^{n -1} < nh(y+1)^{n -1} < x -y^{n -1}$ and so $y$ cannot be $\sup E$ If $y^n > x$, let $k = \frac{y^n -x}{ny^{n -1}}$, then $0< k< y$ and for $t \ge y -k$, we have $\quad\quad y^n - t^n \le y^n -(y -k)^n < nky^{n -1}=y^n -x$ and so $(t\ge y -k)\implies (t^n > x)) $ $\quad\quad y -k (< y)$ is an upper bound of $E$ thus $y\ne \sup E$ Now we see that $y = x^n\quad\quad\square$ Corollary If $a$ and $b$ are positive real numbers and $n$ is a positive integer, $\quad$then $\quad(ab)^{1/n} = a^{1/n} b^{1/n}.$ Proof Put $\alpha = a^{1/n}, \beta = b^{1/n}$, since multiplication is commutative, we $\quad$have $\quad ab = \alpha^n \beta^n = (\alpha\beta)^n.$ Now the uniqueness of $n$th root implies $\quad$that$\quad(ab)^{1/n} = \alpha\beta = a^{1/n} b^{1/n}$ |
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04-26-2012, 02:03 PM
(This post was last modified: 01-29-2019 04:50 PM by elim.)
Post: #13
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1.22 Real Field/Decimals --Rudin [Principle of Math Analysis]
1.22 Decimals Let $0< x\in\mathbb{R}$, let $n_0 = \lfloor x\rfloor$. Notice that the existence of
$n_0$ relies on the archimedean property of $\mathbb{R}$. Having chosen $n_0,\cdots,n_{k-1}$, let $n_k$ be the largest integer such that $\displaystyle{n_0 + \frac{n_1}{10}+\cdots +\frac{n_k}{10^k}\le x} $ and let $\displaystyle{E = \{n_0 + \frac{n_1}{10}+\cdots +\frac{n_k}{10^k} \; | \; k\in \mathbb{N}\}}.$ The decimal expansion of x is $n_0.n_1 n_2 n_3 \cdots.$ (Notice that $0\le n_k < 10\quad(\forall k>0)$) Conversely, for any infinite decimal, the corresponding $E$ is bounded above and the infinite decimal is the decimal expansion of $\sup E$. |
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04-27-2012, 09:35 AM
(This post was last modified: 01-30-2019 01:59 PM by elim.)
Post: #14
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The Extended Real Number System -- Rudin [Principle of Math Analysis]
1.23 Definition The extended real number system consists of the
real field $\mathbb{R}$ and two symbols, $+\infty$ and $-\infty$. Rreserving the original order in $\mathbb{R}$ and define$\;-\infty < x < +\infty\quad (\forall x\in\mathbb{R})$ $\quad\quad$ It's clear that $+\infty$ is an upper bound of every subset of the extended real number system, and every nonempty subset has a least upper bound. We have similar remarks for lower bounds in this system. The extended real number system does not form a field. But we have the following conventions: (a) $\displaystyle{x + \infty = +\infty,\quad x-\infty = -\infty,\quad \frac{x}{+\infty} = \frac{x}{-\infty} = 0\quad (\forall x\in\mathbb{R})}$ (b) $ x\cdot (+\infty) = +\infty,\quad x\cdot (-\infty) = -\infty\quad (0 < x\in\mathbb{R})$ (c) $ x\cdot (+\infty) = - \infty,\quad x\cdot (-\infty) = +\infty\quad (0 > x\in\mathbb{R})$ When it is desired to make the distinction between real numbers on the one hand and the symbols $\pm \infty$ on the other quite explicit, the former are called finite. |
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04-30-2012, 01:08 PM
(This post was last modified: 01-29-2019 06:21 PM by elim.)
Post: #15
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1.24~35 Complex Field -- Rudin [Principle of Math Analysis]
1.24 Definition A complex number is a pair $(a,b)$ of real numbers.
"Ordered" means that $(a,b)$ and $(b,a)$ are regarded as distinct if $a\ne b$ $\quad$ Let $x=(a,b),\; y=(c,d)$ be two complex numbers. We write $x=y$ iff $a=c$ and $b=d$. (This is not entirely superfluous: think of equality of rational numbers, represented as quotients of integers.) We define $\underset{\,}{\qquad}\qquad x+y=(a+c,b+d),\; xy = (ac-bd,ad+bc)$ 1.25 Theorem These definition of addition and multiplication turn the set of all complex numbers $\mathbb{C}$ into a field, with $(0,0)$ and $(1,0)$ in the rule of $0$ and $1$. Proof We simply verify the field axioms in Definition 1.12 using the field structure of $\mathbb{R}$ of course. Let $x = (a,b),\; y=(c,d),\; z=(e,f)$, for example, if $x \ne 0 = (0,0)$, then $a^2+b^2 > 0$ by 1.18d, and we can define $\small\displaystyle{\frac{1}{x} = \left(\frac{a}{a^2+b^2},\frac{-b}{a^2+b^2} \right )}$ since $\small\displaystyle{(a,b)\left(\frac{a}{a^2+b^2},\frac{-b}{a^2+b^2}\right ) = (1,0)}$. That's $(M_5)$ For verify (D) in Definition 1.12, we have $\qquad\begin{align} x(y+z) & = (a,b)(c+e,d+f) \\ & = (ac+ae-bd-bf,ad+af+bc+be) \\ & = (ac-bd,ad+bc)+(ae-bf,af+be) \\ & = xy + xz. \end{align}$ 1.26 Theorem $(a,0)+(b,0) = (a+b,0),\; (a,0)(b,0) = (ab,0).\quad\square$ Theorem 1.26 shows that the complex numbers of the form $(t,0)$ have the same arithmetic properties as the corresponding real numbers $t$. We can therefore identify $(t,0)$ with $t$. This identification gives us the real field as a subfield of the complex field. 1.27 Definition $i = (0,1)$ 1.28 Theorem $i^2 = -1$ Proof $i^2 = (0,1)(0,1) = (-1,0) = -1.\quad\quad\square$ 1.29 Theorem $(a,b) = a + bi\quad (\forall a,b\in\mathbb{R})$ Proof $a+bi = (a,0)+(b,0)(0,1)=(a,0)+(0,b)=(a,b).\quad\square$ 1.30 Definition If $a,b\in\mathbb{R}$ and $z = a+bi$, then $\overline{Z} = a - bi$ is called the conjugate of $z$. The numbers $a$ and $b$ are the real part and the imaginary part of $z$, respectively. $\quad$ We shall occasionally write $a = \operatorname{Re} (z),\quad b = \operatorname{Im}(z).$ 1.31 Theorem If $z$ and $w$ are complex, then $\quad\quad$(a) $\overline{z+w} = \overline{z} + \overline{w}$ $\quad\quad$(b) $\overline{zw} = \overline{z} \overline{w}$ $\quad\quad$(c) $z + \overline{z} = 2\operatorname{Re}(z),\; z -\overline{z} = 2\operatorname{Im}(z)$ $\quad\quad$(d) $z\overline{z}$ is real and positive (except when $z = 0$)$\quad\square$ 1.32 Definition The absolute value $|z|\; (z\in\mathbb{C})$ is a non-negative square root: $|z|=(z\overline{z})^{1/2}$ $\quad$ The existence (and uniqueness) of $|z|$ follows from Th. 1.21&1.31d. $\quad$ Note that when $x \in \mathbb{R}\subset \mathbb{C}$, then $\overline{x} = x$, hence $\qquad\qquad |x| = \sqrt{x^2} = x$ if $x\ge 0$, $|x| = -x$ if $x < 0$ 1.33 Theorem Let $z$ and $w$ be complex numbers. Then $\quad\quad$ (a) $|z| > 0$ unless $z = 0,\; |0| = 0$ $\quad\quad$ (b) $|\overline{z}| = |z|$ $\quad\quad$ (c) $|zw| = |z||w|$ $\quad\quad$ (d) $|\operatorname{Re} z| \le |z|$ $\quad\quad$ (e) $|z+w| < |z|+|w|$ Proof (a) and (b) are trivial. Put $z = a+bi,\; w = c+di,\;(a,b,c,d\in\mathbb{R})$. $\quad\quad$Then$\,|zw|^2 = (ac-bd)^2+(ad+bc)^2 = (a^2+b^2)(c^2+d^2)$ $\quad\quad=|z|^2|w|^2 = (|z||w|)^2$. Now (c) follows from Theorem 1.21. $\quad\quad$ For (d) we have $(a^2 \le a^2+b^2) \implies (|a|\le \sqrt{a^2+b^2})$ $\quad\quad$ For (e) we have $\qquad\begin{align}|z+w|^2 &=(z+w)(\overline{z}+\overline{w})\\ &= z\overline{z}+z\overline{w}+\overline{z}w+w\overline{w}\\ &= |z|^2 + 2\operatorname{Re}(z\overline{w}) + |w|^2\\ &\le |z|^2 + 2|z\overline{w}| + |w|^2\\ &= |z|^2 + 2|z||w| + |w|^2 = (|z|+|w|)^2. \end{align}$ 1.34 Notation If $x_1,\cdots,x_n$ are complex numbers, we write $\qquad\qquad x_1+x_2+\cdots +x_n = {\displaystyle\sum_{j=1}^n} x_j$ We conclude the section with an important inequality, usually known as the Schwarz inequality. 1.35 Theorem If $a_1,\cdots,a_n,\; b_1,\cdots,b_n \in \mathbb{C}$ then $\qquad\qquad\qquad\displaystyle \bigg|\sum_{j=1}^n a_j \overline{b}_j\bigg|^2 \le \sum_{j=1}^n |a_j|^2\sum_{j=1}^n |b_j|^2$ Proof$\quad$ Put $A = \sum |a_j|^2,\; B = \sum |b_j|^2,\; C = \sum a_j\overline{b}_j$ (for this proof, $\quad j = \overline{1,n}$ in all sums). If $B = 0$, then $b_j = 0\quad (j=\overline{1,n})$ and the $\quad$conclusion is trivial. Assume $B > 0$. By Theorem 1.31 we then have $\quad\sum |Ba_j - Cb_j|^2 = \sum (Ba_j - Cb_j)(B\overline{a}_j - \overline{C b_j})$ $\qquad = B^2 \sum |a_j|^2 - B\overline{C}\sum a_j\overline{b}_j - BC\sum \overline{a}_j b_j +|C|^2 \sum |b_j|^2$ $\qquad = B^2 A - B|C|^2 = B(BA - |C|^2)$ $\quad$ Since $B > 0$, we get $AB - |C|^2 \ge 0$. this completes the proof. $\;\;\square$ |
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04-30-2012, 06:03 PM
(This post was last modified: 01-30-2019 02:27 PM by elim.)
Post: #16
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1.36~1.37 EUCLIDEAN SPACES --Rudin [Principle of Analysis]
1.36 Definitions For each positive integer $k$, let $\mathbb{R}^k$ be the set of
all ordered $k$-tuples $\mathbf{x} = (x_1,x_2,\cdots,x_k)$ Where $x_1,\cdots,x_k\in\mathbb{R}$, called the coordinates of $\mathbf{x}$. The elements of $\mathbb{R}^k$ are called points, or vectors, especially when $k>1$. We shall denote vectors by boldfaced letters. If $\mathbf{y} = (y_1,\cdots,y_k)$ and $\alpha \in\mathbb{R}$, put $\qquad\mathbf{x} + \mathbf{y} = (x_1+y_1,\cdots,x_k+y_k), \quad \alpha \mathbf{x} = (\alpha x_1,\cdots,\alpha x_k)$ So $\mathbf{x}+\mathbf{y},\; \alpha \mathbf{x} \in\mathbb{R}^k$. This defines addition of vectors, as well as multiplication of a vector by a real number (a scalar). These two operations satisfy the commutative, associative and distributive laws (trivial) and make $\mathbb{R}^k$ into a vector space over the real field. The zero element of $\mathbb{R}^k$ (sometimes called the origin or the null vector) is the point $\mathbf{0}$ with all the coordinates are $0\underset{\,}{.}$ We shall define the so called "inner product" (or scalar product) of $\mathbf{x}$ and $\mathbf{y}$ by$\;\;\mathbf{x}\cdot \mathbf{y}={\small\displaystyle\sum_{i=1}^k} x_i y_i $ and the norm of $\mathbf{x}$ by $\qquad\qquad\qquad\quad |\mathbf{x}| = (\mathbf{x}\cdot \mathbf{x})^{1/2} = \bigg({\small\displaystyle\sum_{i=1}^k} x_i^2\bigg)$ The structure now defined (the vector space $\mathbb{R}^k$ with the above inner product and norm) is called euclidean $k$-space. 1.37 Theorem Let $\mathbf{x,y,z} \in \mathbb{R}^k$ and $\alpha \in\mathbb{R}$, Then $\quad\quad$ (a) $|\mathbf{x}| \ge 0$ $\quad\quad$ (b) $|\mathbf{x}| = 0 \Longleftrightarrow \mathbf{x} = \mathbf{0} $ $\quad\quad$ (c) $|\alpha \mathbf{x}| = |\alpha||\mathbf{x}|$ $\quad\quad$ (d) $|\mathbf{x}\cdot \mathbf{y}| \le |\mathbf{x}||\mathbf{y}|$ $\quad\quad$ (e) $|\mathbf{x}+\mathbf{y}| \le |\mathbf{x}|+|\mathbf{y}|$ $\quad\quad$ (f) $|\mathbf{x}-\mathbf{z}| \le |\mathbf{x}-\mathbf{y}|+|\mathbf{y}-\mathbf{z}|$ Proof (a),(b) and (c) are obvious and (d) is an immediate $\quad$consequence of the Schwarz inequality. Then by (d) we have $\qquad\begin{align} |\mathbf{x}+\mathbf{y}|^2 & = (\mathbf{x}+\mathbf{y})\cdot (\mathbf{x}+\mathbf{y})\\ & =\mathbf{x\cdot x} + 2\mathbf{x}\cdot \mathbf{y}+\mathbf{y}\cdot\mathbf{y}\\ & \le |\mathbf{x}|^2 +2|\mathbf{x}||\mathbf{y}|+|\mathbf{y}|^2\\ & = (|\mathbf{x}|+|\mathbf{y}|)^2 \end{align}$ $\quad$and so (e) is proved. Finally, (f) follows from (e) when $\mathbf{x,\;y}$ $\quad$are replaced by $\mathbf{x-y,\; y-z}$ respectively. |
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04-30-2012, 06:03 PM
Post: #17
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1.38 Remark(EUCLIDEAN SPACES) --Rudin [Principle of Mathematical Analysis] Notes
1.38 Remarks Theorem 1.37(a),(b) and (f) will allow us (see Chap.2) to regard $\mathbb{R}^k$ as a metric space.
$\quad\quad \mathbb{R}^1$ (the set of all real numbers) is usually called the line, or the real line. Likewise $\mathbb{R}^2$ is called the plane, or the complex plane (compare Definitions 1.24 and 1.36). In these two cases the norm is just the absolute value of the corresponding real or complex number. |
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05-01-2012, 01:25 PM
(This post was last modified: 01-30-2019 05:47 PM by elim.)
Post: #18
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Appendix Ch1 Construction of Real Field --Rudin [Principle of Analysis]
APPENDIX The proof of Theorem 1.19, i.e. constructing $\mathbb{R}$ from $\mathbb{Q}$ will
presented in several steps. Step 1 A set $\alpha$ is called a cut of $\mathbb{Q}$ if $\alpha\subset\mathbb{Q}$ with the following properties: $\quad\quad \begin{align} \text{(I)} & \varnothing \ne \alpha \ne \mathbb{Q}\\ \text{(II)} & (p\in\alpha)\wedge (p> q\in\mathbb{Q}) \implies (q\in\alpha) \\ \text{(III)} & \forall p\in\alpha \exists r\in\alpha\; (p < r) \end{align}$ Let $\mathbb{R} = \{\alpha \mid \alpha \text{ is a cut of } \mathbb{Q}\}$, and assume that in this appendix, the letters $p,q,r,\cdots$ denote rational numbers and $\alpha,\beta,\gamma,\cdots$ denote cuts. $\qquad$ (II) implies two facts which will be used freely: $\qquad\qquad(p\in\alpha)\wedge (q\not\in\alpha)\implies (p < q),\;\;(s > r\not\in\alpha) \implies (s\not\in\alpha)$ The the axioms (M) and (D) of Definition 1.12 hold with $\mathbb{R}^+$ in place of $F$ and $1^*$ in the role of $1$. The proofs are so similar to the ones given in detail in Step 4 that we omit them. Note, in particular, the multiplication property of Definition 1.17 holds: $\alpha,\beta\in 0^* \implies \alpha\beta > 0^*$ Step 2 Define "$\alpha < \beta$" to mean: $\alpha \subsetneq \beta$ ($\alpha$ ia a proper subset of $\beta$) $\quad\quad$ We need to show this meets the requirements of Definition 1.5. $\quad\quad$ Clearly $(\alpha < \beta < \gamma \implies \alpha < \gamma)$ and at most one of $3$ relations $\qquad\;\alpha < \beta, \alpha = \beta, \beta < \alpha$ can hold for any pair of $\alpha,\beta$. To show at $\qquad\;$least one such relation holds, assume that first 2 relations fail, $\qquad\;$then $\alpha$ is not a subset of $\beta$ hence there is a $p\in\alpha$ with $p\not\in\beta$. $\qquad\;$So $q\in\beta \implies q < p \implies q\in\alpha$, i.e. $\beta < \alpha$ $\quad\quad$ Thus $\mathbb{R}$ is now an ordered set. Step 3 $\mathbb{R}$ has the least-upper-bound property. $\qquad$ Let $\varnothing \ne A \subset \mathbb{R}$, and $\beta\in\mathbb{R}$ is an upper bound of $A$. Let $\gamma = \bigcup A$, $\qquad$then $\varnothing \ne \gamma \subset \beta$ hence $\gamma \ne \mathbb{Q}$. It's easy to further prove that $\gamma$ is a cut $\qquad$(a member of $\mathbb{R}$) and an upper bound of $A$. $\qquad$Suppose $\delta < \gamma$, then $s\not\in\delta$ for some $s \in\gamma$. So $s\in\alpha$ for some $\alpha\in A$ $\qquad$and $\delta < \alpha$. Therefore $\delta$ is not an upper bound of $A$. We conclude that $\qquad\gamma = \sup A$. Step 4 Define $\alpha + \beta = \{r+s \mid r\in\alpha,\; s\in\beta \}, \; 0^* = \{r \in\mathbb{Q}: r < 0 \}$. $\quad$Clearly $0^*$ is a cut. We shall verify the axioms for addition (in Definition $\quad$1.12) hold in $\mathbb{R}$ with $0^*$ playing the role of $0$. $\quad(A_1)$ Take ${r}',{s}' \in\mathbb{Q}\setminus (\alpha \cup \beta)$, then $\forall r\in\alpha \; \forall s\in\beta \;(r+s < {r}'+{s}')$ $\qquad\quad\implies ({r}'+{s}' \not\in \alpha+\beta \ne \varnothing)$ So $\alpha+\beta$ has cut property (I). $\quad$ If $p\in\alpha+\beta$ then $p = r+s$ for some $r\in\alpha,\;s \in\beta$. To see (II) holds, $\quad$ for $q < p$, we have $(q-s < r)\implies q-s\in\alpha \implies q = (q-s)+s$ $\quad\;\in \alpha +\beta$; Now choose $t\in\alpha$ so that $t > r$ then $p < t+s \in \alpha+\beta$. $\quad$ Thus (III) holds and so $\alpha + \beta \in\mathbb{R} (\forall \alpha,\beta\in\mathbb{R})$ $\quad(A_2) \quad \alpha +\beta = \{r+s \mid r\in\alpha, s\in\beta\} = \{s+r \mid s\in\beta, r\in\alpha\} = \beta+\alpha$ $\quad(A_3)$ This, similar to the above, follows from the associative law in $\mathbb{Q}$ $\quad(A_4)$ If $r\in\alpha,\; s\in 0^*$, then $r+s < r$ thus $\alpha + 0^*\subset \alpha$. On the other $\quad$hand, if $p\in\alpha$, then $p < r \in\alpha$ for some $r$ so $p = r + (p-r)$ with $p-r$ $\quad\in 0^*$ and so $\alpha \subset \alpha+0^*$. We conclude that $\alpha + 0^* = \alpha$ $\quad(A_5)$ Fix $\alpha\in\mathbb{R}$, let $\beta = \{p\in\mathbb{Q}: \exists r > 0\; (-p -r \in\mathbb{Q}\setminus \alpha)\}$. $\quad$We'll show $\beta\in\mathbb{R}$ and $\alpha+\beta = 0^*.\quad$ If $s\not\in\alpha$, take $p = -s + 1$ then $\quad -p -1 = s\not\in\alpha$ hence $p\in\beta$; If $q\in\alpha$ then $-q \not\in\beta$ so $\varnothing \ne \beta \ne \mathbb{Q}$. $\quad$ For $p\in\beta$, pick $r > 0$ such that $-p-r \not\in\alpha$. If $q < p$, then $-q -r >$ $\quad -p -r$ hence $-q -r \not\in\alpha$ and $q\in\beta$. So (II) holds. Take $t = p + (r/2)$, $\quad$then $-t -(r/2) = -p -r\not\in\alpha$. So $p < t \in\beta$, (III) holds. $\therefore\;\beta\in\mathbb{R}$. $\quad$Let $r\in\alpha,\; s\in\beta$, then$\;-s\not\in\alpha \Rightarrow r < -s \Rightarrow r+s < 0$ so$\small\;\alpha+\beta \subset 0^*$ $\quad$ Conversely, if $v\in 0^*$, let $w = -v/2$ then $w > 0$ and there is an integer $\quad n$ such that $nw\in\alpha$ but $(n+1)w \not\in\alpha$ (by Archimedean property of $\mathbb{Q}$). $\quad$Put $p = -(n+2)w$ then $-p-w = (n+1)w \not\in\alpha \implies p\in\beta$ and $\quad v = -2w = nw + p\in\alpha + \beta$ thus $0^*\subset \alpha+\beta$. $\quad$We conclude that $\alpha + \beta = 0^*$ and got the reason to denote $\beta$ as $-\alpha$. $\quad$(see $(A_4)$) Step 5 Having proved the addition defined in Step 4 satisfied Axioms (A) $\quad$in Definition 1.12, Proposition 1.14 is thus valid in $\mathbb{R}$ and we can prove $\quad$the addition property of ordered field (Definition 1.17): $\quad\quad\quad \forall \alpha,\beta,\gamma \in\mathbb{R}\; (\beta < \gamma)\implies (\alpha + \beta < \alpha + \gamma)$. $\quad$ Indeed, it's clear in this case that $\alpha+\beta \subset \alpha+\gamma$ by the definition of $+$ $\quad$ in $\mathbb{R}$. If $\alpha+\beta = \alpha+\gamma$, then the cancellation law (Proposition 1.14) $\quad$ would imply $\beta = \gamma$. It follows that $\alpha > 0^*$ if and only if $-\alpha < 0^*$. Step 6 Multiplication is a little more bothersome than addition in the $\quad$present context. since products of negative rationals are positive. For $\quad$this reason we confine ourselves first to $\mathbb{R}^+ = \{\alpha\in\mathbb{R}: \alpha > 0^*\}$ $\quad$For $\alpha,\beta \in \mathbb{R}^+$ define $\alpha\beta = \{p\in\mathbb{Q}: p < rs, (0< r\in\alpha,\; 0 < s\in\beta)\}$ $\quad$and $1^* = \{q\in\mathbb{Q}: q < 1\}$ $\quad$The this axioms (M) and (D) of Definition 1.12 hold with $\mathbb{R}^+$ in place $\quad$of $F$ and $1^*$ in the role of $1$. $\quad$The proofs are so similar to the ones given in detail in Step 4 that we $\quad$omit them. Note, in particular, the multiplication property of Definition $\quad$1.17 holds $\alpha,\beta > 0^* \implies \alpha\beta > 0^*$ Step 7 We complete the definition of multiplication by setting $\qquad\qquad\alpha 0^* = 0^* \alpha = 0^*$, and $\qquad\alpha\beta =\begin{cases} (-\alpha)(-\beta)& \text{ if } \alpha < 0^*,\;\beta < 0^* \\ -[(-\alpha)\beta]& \text{ if } \alpha < 0^*,\;\beta > 0^* \\ -[\alpha(-\beta)]& \text{ if } \alpha > 0^*,\;\beta < 0^* \end{cases}$ $\quad$The products on the right were defined in Step 6. $\quad$Having proved (Step 6) that the axioms (M) hold in $\mathbb{R}^+$, it's $\quad$perfectly simple to prove them in $\mathbb{R}$ by repeated application of the $\quad$identity $\gamma = -(-\gamma)$ which is part of Proposition 1.14. $\quad$The distributive law $\alpha(\beta+\gamma) = \alpha \beta + \alpha \gamma$ is proved in several cases. $\quad$For instance, suppose $\alpha > 0^*,\; \beta < 0^*,\; \beta + \gamma > 0^*$, then $\gamma = (\beta+\gamma)$ $\quad\; + (-\beta)$ and (since we already have the distributive law in $\quad\mathbb{R}^+$) $\alpha\gamma = \alpha(\beta+\gamma)+\alpha(-\beta)$. But $\alpha (-\beta) = -(\alpha\beta)$, $\quad$we see that $\alpha(\beta+\gamma) = \alpha \beta + \alpha \gamma$ this case. $\quad$The other cases are handled in the same way. $\quad$We have now completed the proof that $\mathbb{R}$ is an ordered field with $\quad$the least upper-bound property. Step 8 For $r\in\mathbb{Q}$, let $r^* = \{p\in\mathbb{Q}: p < r\}$. It's clear that $r^*$ is a cut; $\qquad$ that is, $r^*\in\mathbb{R}$. We have $\qquad$ (a) $r^* + s^* = (r+s)^*$ $\quad\quad$ (b) $r^* s^* = (rs)^*$ $\quad\quad$ (c) $(r^* < s^*) \Longleftrightarrow (r < s)$ $\quad\quad$ To proof (a), assume $p\in (r^*+s^*)$, then $p = u+v$ for some $\qquad\; u < r,\; v < s$. So $p < r+s$ thus $p \in (r+s)^*$. $\qquad$ Conversely, if $p\in (r+s)^*$, then $p < r+s$. Choose $t$ so that $\qquad\;2t = r+s -p$ and put ${r}' = r-t,\; {s}' = s-t$. Then ${r}'\in r^*,$ $\qquad\;{s}'\in s^*$ and $p = {r}'+{s}'$, so $p\in (r^* + s^*)$. $\quad\quad$ This proves (a). The proof of (b) is similar. $\quad\quad$ (c): $r < s\implies (r\in s^*)\wedge (r\not\in r^*)\implies$ $\qquad\qquad r^* < s^* \implies \exists p\in s^* \; (p\not\in r^*)\implies (r\le p < s)\quad\quad\square$ Step 9 We saw in Step 8 that the correspondence "$r \to r^*$" preserves $\quad$sums, products and order. This can be expressed by saying that the $\quad$ordered field $\mathbb{Q}$ is isomorphic to the ordered field $\mathbb{Q}^*$ whose elements $\quad$are the rational cuts. $\quad$It is this identification of $\mathbb{Q}$ with $\mathbb{Q}^*$ which allows us to regard $\mathbb{Q}$ as a $\quad$subfield of $\mathbb{R}$. The 2nd part of Theorem 1.19 is to be understood in $\quad$terms of this identification. $\quad$It is a fact, which we'll not prove here, that any two ordered fields $\quad$with the least-upper-bound property are isomorphic. The 1st part of $\quad$Theorem 1.19 therefore characterizes the real field $\mathbb{R}$ completely. |
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05-04-2012, 11:17 AM
(This post was last modified: 12-03-2019 05:28 PM by elim.)
Post: #19
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Historical Notes of the Number System Theory --Rudin 【PMA】
$\;\;$The book by Landau and Thurston cited in the Bibliography
are entirely devoted to number systems. Chap. 1 of Knopp's book contains a more leisurely description of how $\mathbb{R}$ can be obtained from $\mathbb{Q}$. Another construction defines real number as an equivalence class of Cauchy sequences of rational num- bers(see Chap.3), is carried out in Section. 5 of the book by Hewitt and Stromberg. $\;\;$The cuts in $\mathbb{Q}$ that we used here were invented by Dedekind. Rational Cauchy sequence equivalence classes' approach of construction is due to Cantor. Both Cantor and Dedekind pub- lished their constructions in 1872. |
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05-04-2012, 11:17 AM
Post: #20
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Chap. I Reader Review --Rudin [Principle of Mathematical Analysis] Notes
Very condensed materials.
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