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If $x < 0$ and $y < z$, then $xy > xz$. +\end{tcolorbox} + +So let's assume the premise. $F$ is an ordered field and $x,y,z \in F$, +and we choose $x,y$ and $z$ such that $x < 0$ and $y < z$. + +From $x < 0$ it follows that $(-x) > 0$. From $y < z$ it follows that $0 < z - y$. +From both of these, we can conclude that $0 < (-x)(z-y)$. +Working out the right side with the distributive law, gives $0 < (-x*z)-(-x*y)$. +Using $-1*-1 = 1$, gives $0 < (-xz)-(-xy)$, thus $0 < xy - xz$. The right part can be split again: $xz < xy$. +Then, the < can be flipped, which gives $xy > xz$. \qed \end{document} -- 2.34.1 From b08ef7f201fcfe5c5ecb6108a9408cf33f0070b4 Mon Sep 17 00:00:00 2001 From: Philippe Zwietering Date: Fri, 19 May 2023 17:14:35 +0200 Subject: [PATCH 2/5] Started on next exercise week 2 --- assignments/week2/main.pdf | Bin 352020 -> 360755 bytes assignments/week2/main.tex | 17 +++++++++++++++-- 2 files changed, 15 insertions(+), 2 deletions(-) diff --git a/assignments/week2/main.pdf b/assignments/week2/main.pdf index 6fe7864352b4406d582e5d725f6da07a01501fb8..7c63797635c4465e91328e94f03d772baac4e9a4 100644 GIT binary patch delta 280622 zcmV(bK>oj!z7?~A7AGZ8L`E$!E;R}z(A34)3NkPtFd%PYY6?6&3NK7$ZfA68ATl;L zG?xKO1t@>r8rhB;$MKzCF>fIivDNqCI7T*rWH`u+pbhe1`C&wImjm-yaY#|}*Sq?d zp6*d|xa%w+I1m;yQ|zj)yQ`{&tNRL9KTh!A`On+Qt2Y`~L?KE9uWs)uMJpu~Qlz?F zRzFYwJJWc&n$6X8_3La-1)kE`FSq{!>^Wnwiu-?MuRpI2Gd*22?aHqr2>#iXlh?Ns 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b/assignments/week2/main.tex index c4a2bd2..3dc243c 100644 --- a/assignments/week2/main.tex +++ b/assignments/week2/main.tex @@ -5,8 +5,6 @@ \exercise*[1.1.1] \begin{tcolorbox} - Prove: - Let $F$ be an ordered field and $x,y,z \in F$. If $x < 0$ and $y < z$, then $xy > xz$. \end{tcolorbox} @@ -19,4 +17,19 @@ Working out the right side with the distributive law, gives $0 < (-x*z)-(-x*y)$. Using $-1*-1 = 1$, gives $0 < (-xz)-(-xy)$, thus $0 < xy - xz$. The right part can be split again: $xz < xy$. Then, the < can be flipped, which gives $xy > xz$. \qed +\exercise[1.1.2] +\begin{tcolorbox} + Let $S$ be an ordered set. Let $A \subset S$ be a non-empty finite subset. Then A is bounded. Furthermore, + $\text{inf} A$ exists and in in $A$ and $\text{sup} A$ exists and is in $A$. +\end{tcolorbox} + +In order to prove that $A$ is bounded, we have to prove that it has an upper and a lower bound. Let us prove +that $A$ is bounded above first. + +In particular, we have to prove that $\exists a \in A$ such that $x \leq b$ for all $x \in E$. +Since $A$ is non-empty and finite, we can use induction on the cardinality of $A$, since that will always +be some natural number $n$. So, we have to prove two cases: the base case, where $|A|=1$, and the inductive step, +where we will assume that when $A$ has an upper bound when it has cardinality $m$, +then it also has an upper bound when its carindality is equal to $m+1$. + \end{document} -- 2.34.1 From e36d1e473385eac732ce4e2f8ad4e651526e9b77 Mon Sep 17 00:00:00 2001 From: Philippe Zwietering Date: Mon, 22 May 2023 07:31:35 +0200 Subject: [PATCH 3/5] Another exercise done week 2 --- assignments/week2/main.pdf | Bin 360755 -> 364797 bytes assignments/week2/main.tex | 21 ++++++++++++++++++++- 2 files changed, 20 insertions(+), 1 deletion(-) diff --git a/assignments/week2/main.pdf b/assignments/week2/main.pdf index 7c63797635c4465e91328e94f03d772baac4e9a4..7f7ccc16b8eab168ac8c531a0e8d02f11d30b409 100644 GIT binary patch delta 294310 zcmV)NK)1iMf)@Ru7APf9L`E$!E;R}z(A34)3NkeyFd%PYY6?6&3NK7$ZfA68AT%;L zI3SY|3nzc=9BXeQxAD7wg}zlzt#y$& z-&}Kdt>lWJMQ|{rr8wlwaNeA~cJ?r9XJ22nz0WuAufF_>5IrNRp+xZ6`}>)qGbL1` 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zn@~|6zWI5bFYzrnbmnB<|5uT|^cRwRHF+X~EEtw7LpK2y4`4S&V$-zuwAaBIo#br{Ut_NzTJZ&Q7j@#3ujM#)X`R|NjS4&|&vUf==anbKWm?>f0f5;uCeJF;^kdR0R z&7mL>B%p@P$YIRoD&^K|V#RRQXG2LJ%pr^kBtV?AVa(hV@TjAbxJYW@MG6;3wP{+q ziW%c)!ep$<+KW!hPpvgBYh5gFbw{qvB+s>+VT6+vClZ%A|l82|cyQi Date: Tue, 23 May 2023 17:04:40 +0200 Subject: [PATCH 4/5] Finished a few more exercies of week 2 --- assignments/main_text.pdf | Bin 580687 -> 597422 bytes assignments/week2/main.pdf | Bin 364797 -> 381025 bytes assignments/week2/main.tex | 85 ++++++++++++++++++++++++++++++++++++- 3 files changed, 83 insertions(+), 2 deletions(-) diff --git a/assignments/main_text.pdf b/assignments/main_text.pdf index 4760256363cd0b7abe02176ff781d9aa5e5bf70d..df91bf2a1b031f8370da607659c847e46c23ff62 100644 GIT binary patch delta 395489 zcmV)UK(N2hxFfE+BoQT0L`E$!E;R}z(A34)3O6{B5fcJ5HkZM=2q}M^lS@kjF%ZY^ z{S-MShm13`c^kJ&!edM^i2Om^%8N6lGcR+c*`mBu+RE{?SzH&hMKe28#Bw=X zyesl@Q*P8W3v8? z{Ra~}#`I#Ur?88^q<=S^e=^+DmWo$Igf4BuRH 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zo~&ntAg_Xc6%1?V5CXh^s$>pdg1Oi_a(vVJ{ONa;pZ?{(vDtRF*4xrvEHw3T+fY|5 z!M585U&i!Zy`BLyfbE~}=&_p$wGvHft8m~Sbk5vXJMoBV<4B{U2yVMQLAuU5w0 z&fD^o{<-+QfI_fYf&lbLA!)_vB2@ge`5G&<$q@)GYQIYCxYp~iEn|4U{8O!qTl#KQ z_%b#!B{}?#S-!_-^q?&3qbYp?*D8_fMAN@TM+U^Id^pO&fgUAd)!%MO@mQ_>c~6ej zPv8*vz{|utu7SvDoSy#3scy=AQ28g>vTBxnht+Sw>8ZNCaJh8wk!`*%{RF%!9X2KX zTmRQDEqfF_O*GT#mxS55yTkkHSuZm~BtGI|t`Rt~uvpD0!(4T%aDGF#ucKz)>R0HQxL=eJEtxzgNmh> zDKR@IEQ1E|PbOj(Vy3U0vc0_vF*_SEBe60pgOr__Juy4`e*;nCpS(h>j4Yf&V(g+q zY)tIz;_OV!B5Z8pj9kKO9HLx|LM;5m|NoY+Q{etvIx}&y68~@MOfcPU3#Nn~wClX2 zbI)GQ208IoYX)ju?LUJ+Y?TX24uuE}9rrzc{u@Oaszrd=&OeS9)?Ud16qHs^(&BIY zyn!WL6~Rft3W_&7F#J29Msa~O*d9T=9U}OsaY5+lYUIc4b8-Hl-x!0C#cv=XutlK8 zW xz$. \qed \exercise[1.1.2] \begin{tcolorbox} Let $S$ be an ordered set. Let $A \subset S$ be a non-empty finite subset. Then A is bounded. Furthermore, - $\text{inf} A$ exists and in in $A$ and $\text{sup} A$ exists and is in $A$. + $\text{inf} A$ exists and in in $A$ and $\sup A$ exists and is in $A$. \end{tcolorbox} In order to prove that $A$ is bounded, we have to prove that it has an upper and a lower bound. Let us prove @@ -48,7 +48,88 @@ it must also be bounded.}. This will be left to the reader. \exercise[1.1.5] \begin{tcolorbox} Let $S$ be an ordered set. Let $A \subset S$ and suppose $b$ is an upper bound for $A$. Suppose $b \in A$. - Show that $b = \text{sup} A$. + Show that $b = \sup A$. \end{tcolorbox} +So, let $S$ be an ordered set, with $A \subset S$ and $b \in A$ being an upper bound for $A$. Since $b$ is an upper +bound, $a \leq b$ for all $a \in A$. Since $b \in A$ as well, we know that there is some element in $A$ which is +the greatest element of them all, and all other elements are smaller. + +Now let's assume that $b \neq \sup A$. Then either some other element of $A$ is the supremum, +which would imply that $b$ is not larger than this element, which is a contradiction. +The other possibility is that there is an element $c \in S\backslash A$ that is the +supremum. Because $S$ is ordered, $c$ must either be greater than, smaller than or equal to $b$. If $c < b$, c is +not an upper bound of $A$ and thus also not its supremum. If $c > b$, then $b$ is an upper bound that is smaller than +$c$ and therefore $c$ cannot be the supremum. The only option left is that $c=b$, and therefore $b = \sup A$. +\qed + +\exercise[1.1.6] +\begin{tcolorbox} + Let $S$ be an ordered set. Let $A \subset S$ be nonempty and bounded above. Suppose $\sup A$ exists + and $\sup A \notin A$. Show that $A$ contains a countably infinite subset. +\end{tcolorbox} + +Let $S$ be an ordered set, with $A \subset S$ nonempty and bounded above. We assume that $b = \sup A$ exists and +$b \notin A$ $(\implies b \in S\backslash A)$. We are asked to show this then implies that $\exists X \subset A$ +such that $|X| \geq |\N|$. We will prove this with a proof by contradiction. + +We assume that no such set $X$ exists, i.e. $|X| < |\N|$. So, $A$ also doesn't have to countably infinite anymore. +Since $b \notin A$ and $A$ is ordered, finite and nonempty, there is a greatest element $a \in A$ such that +$a < b$ and $x < a $ $\forall x \in A$. So $b$ is in fact not the least upper bound of $A$, +which is in contradiction with our assumption earlier. Ergo, $|X| \geq |\N|$. Since $X$ then is at least of the same +cardinality as $\N$, it must also contain a countably infinite subset. \qed + +\exercise[1.2.7] +\begin{tcolorbox} + Prove the arithmetic-geometric mean inequality. That is, for two positive real numbers $x, y$, we have + \begin{equation*} + \sqrt{xy} \leq \frac{x + y}{2}. + \end{equation*} + Furthermore, equality occurs if and only if $x = y$. +\end{tcolorbox} + +Let us prove the first statement first. So we let $x,y \in \R$ such that $x,y > 0$. Then we will prove the statement +by contradiction. Hence, we assume that +\begin{equation*} + \sqrt{xy} > \frac{x + y}{2}. +\end{equation*} + +We can multiply both sides with 2. This results in $2\sqrt{xy} > x + y$. We can pull the left part into the right, +so we get $0 > x - 2\sqrt{xy} + y$. We can restructure the right side to $0 > (\sqrt{x} - \sqrt{y})^2$. +We know that $0 \leq z^2$, $\forall z \in \R$, so this is a contradiction. \qed + +Now to prove the second statement. We assume $x = y$ is a positive real number. Then, +\begin{equation*} + \sqrt{xy}=\sqrt{x^2}=x=\frac{2x}{2}=\frac{x + y}{2}. \qed +\end{equation*} + +\exercise[1.2.9] +\begin{tcolorbox} + Let $A$ and $B$ be two nonempty bounded sets of real numbers. Define the set $C := \{a + b : a \in A, + b \in B\}$. Show that $C$ is a bounded set and that + \begin{align*} + \sup C &= \sup A+\sup B \; \text{and}\\ + \inf C &= \inf A+\inf B. + \end{align*} +\end{tcolorbox} + +First, let us show that $C$ is a bounded set. Since $A$ and $B$ are both subsets of $\R$, which is an ordered field, +all elements of $C$ must also be real numbers. Let $a$ be an upper bound for $A$ and $b$ be an upper bound for $B$. +So $x \leq a \; \forall x \in A$ and $y \leq b \; \forall y \in B$. Since $C$ is defined as the sum of any element in +$A$ with any element in $B$, an upper bound of $C$, $c$, can be found as $c \leq a + b$. A similar argument can be +made for the lower bound of $C$, which makes $C$ bounded. \qed + +To prove that $\sup C = \sup A + \sup B$, we will show that $\sup C \geq \sup A + \sup B$ and +$\sup C \leq \sup A + \sup B$. + +Let $a = \sup A$ and $b = \sup B$. So $x \leq a$ for all $x \in A$ and $y \leq b$ for all $y \in B$. Then, +$x + y \leq a + b$. Since $z \leq x + y$ for all $z \in C$ because of the definition of $C$, +$z \leq a + b$. In other words, $\sup C \leq \sup A + \sup B$. + +Now to prove the other direction. Let $c = \sup C$. So $z \leq c$ for all $c \in C$. Since all elements in $C$ are +the sum of an element $x \in A$ and $y \in B$, $x + y \leq c$ for all $x,y$. The least upper bound for these $x$ and +$y$ can be given by the supremum; $x \leq \sup A$ and $y \leq \sup B$. So, $\sup A + \sup B \leq c \implies +\sup A + \sup B \leq \sup C$, completing the equality. \qed + +A similar argument can be given for the infimum, which is left to the reader. \end{document} -- 2.34.1 From 73d1b0e3211fe83ecbc1e98a5eb1a990a7ded3da Mon Sep 17 00:00:00 2001 From: Philippe Zwietering Date: Thu, 25 May 2023 16:33:49 +0200 Subject: [PATCH 5/5] Finished assigment 2 --- assignments/main_text.pdf | Bin 597422 -> 602056 bytes assignments/week1/main.pdf | Bin 487434 -> 487932 bytes assignments/week1/main.tex | 2 +- assignments/week2/main.pdf | Bin 381025 -> 469628 bytes assignments/week2/main.tex | 55 +++++++++++++++++++++++++++++++++++++ 5 files changed, 56 insertions(+), 1 deletion(-) diff --git a/assignments/main_text.pdf b/assignments/main_text.pdf index df91bf2a1b031f8370da607659c847e46c23ff62..90488684abcb65f4db9b5504d513d4ed2d99c1c4 100644 GIT binary patch delta 104307 zcmV)BK*PVTyClfpBoQT0L`E$!E;R}z(A34)3OF;75fcJ6GM8cF2r7S+TT8=05QX3S 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Since $\N$ is countably infinite, Each set is definitely finite, because they all contain just one element. Finally, the union of the collection of sets is equal to $\N$, which is not a finite set. -\exercise* +\exercise*[6] \begin{tcolorbox} \begin{enumerate}[label=\emph{\alph*)}, wide] \item Compute $f(4/15)$. 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zx&WJHtDt2gu1_x(lgygf0NYT>%5cdnl`9RjKMROP^^9{iR$3_bC-rRI!ny$8=0p 0 \; \text{and} \; x^3 < 2\}. + \end{equation*} + + \begin{enumerate}[label=\emph{\alph*)}] + \item Prove that $E$ is bounded above. + \item Let $r = \sup E$ (which exists by part a)). Prove that $r > 0$ and $r^3=2$. + + \textit{Hint:} Adapt the proof used in Example 1.2.3. + \end{enumerate} +\end{tcolorbox} + +So, let $E$ and $r$ be defined as in the exercise statement. Then: +\begin{enumerate}[label=\emph{\alph*)}] + \item $x \leq 2$ is an upper bound for $E$, as $2 * 2 * 2 = 8$. So, $E$ is bounded above. + \item As $1 \in E$, $r \geq 1 > 0$, so the first part of the statement holds. In order to show that $r^3=2$, + we want to show that $r^3 \leq 2$ and $r^3 \geq 2$ hold. + + First, let's show that $r^3 \geq 2$. We will take a similar approach as in Example 1.2.3 from the textbook. + So, take a positive number $s$ such that $s^3 < 2$. We wish to find an $h > 0$ such that $(s + h)^3 < 2$. + As $2 - s^3 > 0$, we have $\frac{2 - s^3}{3s^2 + 3s + 1} > 0$. Choose an $h \in \R$ such that $0 < h < + \frac{2 - s^3}{3s^2+3s+1}$. Furthermore, assume $h < 1$. Estimate, + + \begin{align*} + (s+h)^3 - s^3 &= h(3s^2 + 3sh + h^2) \\ + &< h(3s^2 + 3s + 1) \quad &\text{(since } h < 1) \\ + &< 2 - s^3 \quad &\text{(since } h < \frac{2 - s^3}{3s^2 + 3s + 1}). + \end{align*} + + Therefore, $(s+h)^3 < 2$. Hence $s + h \in E$, but as $h > 0$, we have $s + h > s$. So $s < r = \sup E$. + As $s$ was an arbitrary positive number such that $s^3 < 2$, it follows that $r^3 \geq 2$. + + Now take an arbitrary positive number $s$ such that $s^3 > 2$. We wish to find an $h > 0$ such that + $(s-h)^3 > 2$ and $s - h$ is still positive. As $s^3 - 2 > 0$, we have that $\frac{s^3 - 2}{3s^2+1} > 0$. + Let $h := \frac{s^3-2}{3s^2+1}$, and check that $s - h = s - \frac{s^3-2}{3s^2+1} = + \frac{2s^3+s+2}{3s^2+1} > 0$. Assume that $h < 1$. Estimate, + + \begin{align*} + s^3 - (s-h)^3 &= h(3s^2 - 3sh + h^2) \\ + &< h(3s^2+h^2) \quad &\text{(since } s>0 \; \text{ and } h > 0) \\ + &< h(3s^2+1) \quad &\text{(since } h < 1) \\ + &= s^3 - 2 &\text{(because of the definition of } h). + \end{align*} + + By subtracting $s^3$ from both sides and multiplying by -1, we find $(s-h)^3>2$. Therefore, $s-h \notin E$. + Moreover, if $x \geq s-h$, then $x^3 \geq (s-h)^3 > 2$ (as $x > 0$ and $s-h > 0$) and so $x \notin E$. Thus, + $s - h$ is an upper bound for $E$. However, $s-h < s$, or in other words, $s > r = \sup E$. + Hence, $r^3 \leq 2$. + + Together, $r^3 \geq 2$ and $r^3 \leq 2$ imply $r^3 = 3$. \qed +\end{enumerate} \end{document} -- 2.34.1

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