From 8d1cac7611d321ba68a145c6f49e472a5ebb2889 Mon Sep 17 00:00:00 2001 From: Philippe Zwietering Date: Tue, 23 May 2023 17:04:40 +0200 Subject: [PATCH] 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 zcje}Qpha*nq@_6I%y8bEy>|96YiD0ywf^(X`>QX%(riXlLy6$C_xCeJXG*9>k=gsZ z+4uABu5~+KUoX^r{m<(K744i}|Ly&s0D8fH7|h~s+S?!3PuF_B+IDLK4SpOC@dV{;=pQl#kp!)YB)=yI~+9)Q%Yrj zbGaZ5)r8Zyel`*?zy;%I*lw2(2BgvsmZFFSKH(}J79RmVK;a|)<b1V#s|3E<1UN@IVS=zf)Cm#5M&T_ z;?K^VU;4hZx1YTU{FWGSOEUO2&-*j8fx??fgoaxgc%LsYv}x& z`zpG!VUlI&GOp&otKml5Mw`gCwze%uDUXN^@CTMLI^y;TH*8_U1ywrmq+AsAg^RfF zjbr|4vp&>TXKf=nDY@_W_=DSj!;MvUVg`27iEU`w`*NF^`kqXJ;C`F*W8oEKku>C6 zt5t_tF%1N9kvWWt=p(-}=OSS@1{$yj6yF_zCMM^*_32j~epT~j*PZukHw;X2UmmR> z@6mICtT$CT@LSLp50RJqRAnr~i-OLzeV}dUHn%zjm_?m3s8ibN6#Sfjbjr@X8$*tP zEJ!BGeSRht%gAR6!UUsAv`|pFvl5oHXJ<>06RXV;jmi;@V}w``dhpG9l-O~+xS|c0 zq+DTw#T=wrgFz@@{ynjgXo0D+$AIp<+SoWM4~@W)5ilZl!fCUt8gL3&^Zy--)VN@= z)KZutDEHcff`8nPaovD_&E+69|zp^ZVm&cf3~f=gvK6 zry^>o5+j?IlxsC#REY!mZ8O(WSrpc2<57QETLf&ZQenm)s)#|fA@!xapiF~3?&FV# zZ7w+|s*tE2_yr=9TZ^`Z0FZv3(_j)LT=kB&tx=#G9hp>O+FKcaINnC#{-VB=E04%0HGl1YP>M#5mckh2rUd?iREdL- zJyU{=Ix$EctYl5;@7)KP@;;I`9Mqz~#zu<73aXh9R0?pkH9wcMW|>oOF&C=7R@i7i zQL8P@U} zsA)slNO2yD>Z=jpW<)g!IEDc-&vxn_{vels#vr)ouhx?nuPK?;3Nzq zB9*d5!;RFbMj8i-l5iX?>gx$O)y=~O*-*gYf4ab)b25(6MiX=msgw;;VIU`u+zbOH zH^XRuQD9HNscs%N$cA!m{y!AhIr|(1c5al7TO@U6G80RL2YV85p7Tm5X zLEhCYWws3#%CYKM041BQn5!H@1W~KT!*I7gfwy3aPGLtO*x*8zJKdsjXIKDK_>E5D zOC?53W9T}x4#&e{y?;D?a*tTwJZw*XPIvx)n9xQi=O*2{Uv0@09(^i&KHNEao5F?r z^9kqaYE0)R@RLvbqhpd!m%)l|57n~2Zb3d{#Bp0C9V|e)8CYMd`Hzrob?NfMuOB~a z)f?Gpu3oWCD$$7-CAt_bcIU_HaT|z@_4HkyNpzttg8E&So)c#uNGYf*ki*=f&L*mz$%xdg@NGsGW-p zBM>)|Tx6)tAGWLWZuxXAz0tu#F_l9J5=ARiPnYH>$LQ6RSbSius#hRXG8fBwNz4%W zs#~Bb>7^j@g!I^CFG2eB4c~3a9)0y93`8#}+j_1iWO;J`aRPHDz!SrY#n$(KM{fp} zyWJ?>Xuhz4q=I+J@&fHKmPZ+XJApVStdWpF6npzBe|e9~V2w^|&a1@w=U>vM|s^VoD$%aCDaM9xZEU;oI{Tgvfqd?w*&Q zI`=(APe5aL+d2S1qJ9-!I9{OwB>eIss!G+ zhojQzsH8P>{y{l22cM64*PWX={O#GAJ`1#3cI)mpPF(#F%K67-=cnR-&-rc}9|3!l zz^lipTwN_k{R5NLDi+LOee2d+wg_ahb$W$n6)ow7bx^z%!aLU-O;4uR4HY<|g3t#) z0wdjN{pdIoWP}AFA&n3kjFM|eunR}Vl>q|}Qi=XObWI2zqXUM3hta%(QF}XC>^7)7 zEIKjfYAm60m-Q>G1*U{&n35!T>*6yShx1^ zlS7Mxx_8HZK!oiQ!svqX)OuA=-u5VOBjpgChq(QP(aS=38}R&aUhaHob*w4^L}=hj zf(-8m%H^0s`YFGL2RYx~kIN`&v{B`Lw*wT)RHg<+X2AHrSJxH|%R4j%t}P=E{xO+Z zgHt6v98shs8*KC<*=;(em<8;Q6B`0$jX=rIfdZdT;V>So)mWa^-N_BCckBK7aB|b8 z`e?X9vsh#MAX;31XgDif0yzbdK|xHWATJ#xV~{y%o);%vP69qg&Hv)E7OS8>yknuf z&XSpprZSjW5B}Y?rs1b#Fj;0W7TQnkZVUuMCCts(E}{k7i>{-8BzmIF3Tpv<)V5u)Fob4; zK$*Rf>j^^4|Gu&t*3~xU|Z$%sXP6e4S??PF@Js;M= z)+D<*CNOw21YWb)Z&S}ASxW6M$|8qMN2HT-ZW{@U&Fi5;5^!ZZ9Se{*lN&+oq^K&P zZXClxrJ$V_P)tvls`6@hZ=2}#5Drliw$VV^c5;}1+=s3#dz>yrCb?AFVJFgepWLrg zQY!6r;YHliS;+}9BgjsRtU0y+uu9}dC(>O4`H>W?TzC=_qOoaEmQX3{w~{`3|G*jv zzNr8oj50YbqD(TJOqtBuwh%uR>-&T?UZhi>nU4w<7!%PW^WD_$jB+r~qRt$fZG|lB zMF;qQO*YB5>xBaB>bQSgx|w6nT2mXA7kh7}Fh&g{+;t5(R9B#Ve@-syn#? 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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}

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