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\documentclass[12pt]{amsart} \usepackage{amssymb} \usepackage{mathabx} \usepackage{bm} \usepackage{wasysym} \usepackage{graphicx} \usepackage{stmaryrd}
\usepackage{imakeidx}
\usepackage{color,hyperref}
\hypersetup{colorlinks,breaklinks,
             linkcolor=blue,urlcolor=blue,
           anchorcolor=blue,citecolor=blue}
\newenvironment{pf}{\proof[\proofname]}{\endproof}

\makeindex[name=terms,title=Index of terms, columns=2]

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\renewcommand\theequation{\arabic{section}.\arabic{equation}}

\newcommand{\wrlab}[1]{\label{#1}}
\renewcommand{\arraystretch}{1}

\newtheorem{thm}{THEOREM}[section]
\numberwithin{thm}{subsection}
\newtheorem{axiom}[thm]{AXIOM}
\newtheorem{defn}[thm]{DEFINITION}

\newcommand{\secref}[1]{\S\ref{#1}}

\newcommand{\tref}[1]{Theorem~\ref{#1}}
\newcommand{\aref}[1]{Axiom~\ref{#1}}
\newcommand{\dref}[1]{Definition~\ref{#1}}

% BLACK BOARD BOLD

\def\Bbb{\mathbb}  %needed in Nancy

\def\A{{\mathbb A}} \def\B{{\mathbb B}} \def\C{{\mathbb C}} \def\D{{\mathbb D}}
\def\N{{\mathbb N}} \def\I{{\mathbb I}} \def\Q{{\mathbb Q}} \def\R{{\mathbb R}}
\def\Z{{\mathbb Z}} 

\def\T{{\mathbb T}}

\def\P{{\mathbb P}} 

% BOLD
\def\bfzero{{\bf 0}}
\def\bfP{{\bf P}}
\def\bfQ{{\bf Q}}
\def\bfS{{\bf S}}
\def\bfT{{\bf T}}

% CAL 
\def\Cal{\mathcal} \def\scra{{\mathcal A}} \def\scrb{{\mathcal B}}
\def\scrd{{\mathcal D}} \def\scrf{{\mathcal F}} \def\scrg{{\mathcal G}}
\def\scri{{\mathcal I}} \def\scrk{{\mathcal K}} \def\scrl{{\mathcal L}}
\def\scrm{{\mathcal M}} \def\scrp{{\mathcal P}} \def\scrq{{\mathcal Q}}
\def\scrr{{\mathcal R}} \def\scrs{{\mathcal S}} \def\scrt{{\mathcal T}}
\def\scru{{\mathcal U}} \def\scrv{{\mathcal V}} \def\scrw{{\mathcal W}}
\def\scre{{\mathcal E}} \def\scrx{{\mathcal X}} \def\scry{{\mathcal Y}}
\def\scrz{{\mathcal Z}} \def\scrh{{\mathcal H}} \def\scrc{{\mathcal C}}
\def\scrn{{\mathcal N}} \def\scro{{\mathcal O}}
\def\scrpf{{\scrp\scrf}}

% HATTED ROMAN
\def\hatd{{\widehat{d}}}
\def\hatX{{\widehat{X}}}

% HATTEDCAL
\def\hatscrp{{\widehat{\scrp}}}
\def\hatscrp{{\widehat{\scrp}}}
\def\hatpi{{\widehat{\pi}}}

% HATTED BB BOLD
\def\hatbbR{{\widehat{\R}}}

\newcommand\scrlito{
  \mathchoice
    {{\scriptstyle\mathcal{O}}}% \displaystyle
    {{\scriptstyle\mathcal{O}}}% \textstyle
    {{\scriptscriptstyle\mathcal{O}}}% \scriptstyle
    {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}}%\scriptscriptstyle
  }

\def\scrbigo{{\widehat{\scro}}}

% LIE ALGEBRAS

\font\tenolde=eufm10 at 10pt
\font\sevenolde=eufm7
\font\fiveolde=eufm5
\newfam\oldefam
\textfont\oldefam=\tenolde
\scriptfont\oldefam=\sevenolde
\scriptscriptfont\oldefam=\fiveolde
\def\fraktwo{\fam\oldefam\tenolde}

\newfam\cyrfam
\font\tencyr=wncyr10
\def\cyr{\fam\cyrfam\tencyr}

\newcommand\sbullet[1][.75]{\mathbin{\vcenter{\hbox{\scalebox{#1}{$\bullet$}}}}}

\def\aff{{\mathfrak{aff}(\R)}} \def\La{{\mathfrak a}}
\def\Lb{{\mathfrak b}} \def\Lc{{\mathfrak c}} \def\Lg{{\mathfrak g}}
\def\Lh{{\mathfrak h}} \def\Lk{{\mathfrak k}} \def\Ll{{\mathfrak l}}
\def\Lm{{\mathfrak m}} \def\Ln{{\mathfrak n}} \def\Lr{{\mathfrak r}}
\def\Ls{{\mathfrak s}} \def\Lt{{\mathfrak t}} \def\Lw{{\mathfrak w}}
\def\Lq{{\mathfrak q}} \def\Lu{{\mathfrak u}} \def\Lv{{\mathfrak v}}
\def\Lz{{\mathfrak z}} \def\Lsl{\mathfrak{sl}} \def\Lgl{\mathfrak{gl}}
\def\Lp{{\mathfrak p}} \def\Lf{{\mathfrak f}} \def\Lx{{\mathfrak x}}
\def\Lpsl{\mathfrak{psl}}
\def\Lso{{\mathfrak{so}}} 
\def\Lsu{{\mathfrak{su}}} 
\def\Lsp{{\mathfrak{sp}}} 
\def\orthog{{\mathfrak{o}}} 
\def\tilso{{\mathfrak{so}}}
\def\tilco{{\mathfrak{co}}} 
\def\co{\mathfrak{co}}

% OVERLINE
\def\barscru{{\overline{\scru}}}
\def\barPsi{{\overline{\Psi}}}
\def\barA{{\overline{A}}}
\def\barB{{\overline{B}}}
\def\barC{{\overline{C}}}
\def\barD{{\overline{D}}}
\def\barE{{\overline{E}}}
\def\barF{{\overline{F}}}
\def\barbarF{{\overline{\overline{F}}}}
\def\barG{{\overline{G}}}
\def\barH{{\overline{H}}}
\def\barI{{\overline{I}}}
\def\barJ{{\overline{J}}}
\def\barK{{\overline{K}}}
\def\barL{{\overline{L}}}
\def\barM{{\overline{M}}}
\def\barN{{\overline{N}}}
\def\barP{{\overline{P}}}
\def\barbarP{{\overline{\overline{P}}}}
\def\barQ{{\overline{Q}}}
\def\barR{{\overline{R}}}
\def\barS{{\overline{S}}}
\def\barT{{\overline{T}}}
\def\barU{{\overline{U}}}
\def\barV{{\overline{V}}}
\def\barW{{\overline{W}}}
\def\barbarW{{\overline{\overline{W}}}}
\def\barX{{\overline{X}}}
\def\barY{{\overline{Y}}}
\def\barZ{{\overline{Z}}}
\def\barLr{{\overline{\Lr}}}
\def\barLg{{\overline{\Lg}}}
\def\barLh{{\overline{\Lh}}}
\def\barLp{{\overline{\Lp}}}
\def\barLn{{\overline{\Ln}}}
\def\barLm{{\overline{\Lm}}}
\def\barLl{{\overline{\Ll}}}
\def\bara{{\overline{a}}}
\def\bare{{\overline{e}}}
\def\barf{{\overline{f}}}
\def\barg{{\overline{g}}}
\def\bark{{\overline{k}}}
\def\baru{{\overline{U}}}
\def\barv{{\overline{v}}}
\def\barw{{\overline{w}}}
\def\barz{{\overline{z}}}
\def\barbeta{{\overline{\beta}}}
\def\bargamma{{\overline{\gamma}}}
\def\barI{{\overline{I}}}

% TILDE
\def\tilvec{\widetilde{\vec}}

% HAT
\def\hatA{{\widehat{A}}}
\def\hatC{{\widehat{C}}}
\def\hatH{{\widehat{H}}}
\def\hatI{{\widehat{I}}}
\def\hatN{{\widehat{N}}}
\def\hatQ{{\widehat{Q}}}
\def\hatR{{\widehat{R}}}
\def\hatS{{\widehat{S}}}
\def\hatV{{\widehat{V}}}
\def\hatt{{\widehat{t}}}
\def\hatu{{\widehat{u}}}
\def\hatv{{\widehat{v}}}
\def\hatw{{\widehat{w}}}
\def\hatx{{\widehat{x}}}
\def\haty{{\widehat{y}}}
\def\hatz{{\widehat{z}}}
\def\hatalpha{{\widehat{\alpha}}}
\def\hatrho{{\widehat{\rho}}}
\def\hatkappa{{\widehat{\kappa}}}
\def\hatlambda{{\widehat{\lambda}}}
\def\hatsigma{{\widehat{\sigma}}}
\def\hatmu{{\widehat{\mu}}}
\def\hatnu{{\widehat{\nu}}}
\def\hattau{{\widehat{\tau}}}

% CHECK
\def\ckt{{\widecheck{t}}}
\def\ckw{{\widecheck{w}}}
\def\ckx{{\widecheck{x}}}
\def\cky{{\widecheck{y}}}
\def\ckz{{\widecheck{z}}}
\def\cksigma{{\widecheck{\sigma}}}
\def\cktau{{\widecheck{\tau}}}
\def\ckH{{\widecheck{H}}}
\def\ckN{{\widecheck{N}}}
\def\ckS{{\widecheck{S}}}

\def\ds{\displaystyle}
\def\bfindex[#1]{{\bf#1}\index[terms]{#1}}

\def\timestimes{{\times\!\!\times}}

%operators
\def\PF{{\rm PF}}
\def\set{{\rm set}}
\def\bi{{\rm bi}}
\def\fn{{\rm fn}}
\def\ss{{\rm setset}}
\def\sf{{\rm setfn}}
\def\fs{{\rm fnset}}
\def\seq{{\rm seq}}
\def\EUCL{{\rm EUCL}}
\def\TNSR{{\rm TNSR}}
\def\starstar{{**}}
\def\fincap{{\hbox{{\rm fin}}\,\cap}}
\def\fincup{{\hbox{{\rm fin}}\,\cup}}
\def\ctblcap{{\hbox{{\rm ctbl}}\,\cap}}
\def\ctblcup{{\hbox{{\rm ctbl}}\,\cup}}
\def\finamalg{{\hbox{{\rm fin}}\,\amalg}}
\def\two{{\rm two}}
\def\three{{\rm three}}
\def\ten{{\rm ten}}
\def\Lin{{\rm Lin}}
\def\Bilin{{\rm Bilin}}
\def\Trilin{{\rm Trilin}}
\def\PS{{\rm PS}}
\def\UE{{\rm UE}}
\def\FV{{\rm FV}}
\def\CH{{\rm CH}}
\def\DQ{{\rm DQ}}
\def\SS{{\rm SS}}
\def\SUMS{{\rm SUMS}}
\def\res{{\rm res}}
\def\add{{\rm add\,}}
\def\FS{{\rm FS}}
\def\LIMS{{\rm LIMS}}
\def\NOT{{\rm NOT}}
\def\ORB{{\rm ORB}}
\def\dlin{{\rm dlin}}
\def\LINS{{\rm LINS}}
\def\opLIMS{\mathop{\LIMS}}
\def\IDS{{\rm IDS}}
\def\UL{{\rm UL}}
\def\UB{{\rm UB}}
\def\LB{{\rm LB}}
\def\LP{{\rm LP}}
\def\LPD{{\rm LPD}}
\def\Isol{{\rm Isol}}
\def\Cl{{\rm Cl}}
\def\sCl{{\rm sCl}}
\def\id{{\rm id}}
\def\dim{{\rm dim\,}}
\def\DIMS{{\rm DIMS\,}}
\def\Bas{{\rm Bas\,}}
\def\Int{{\rm Int}}
\def\IntD{{\rm IntD}}
\def\det{{\rm det\,}}
\def\exp{{\rm exp}}
\def\End{{\rm End}}
\def\SO{{\rm SO}}
\def\CO{{\rm CO}}
\def\O{{\rm O}}
\def\Stab{{\rm Stab}}
\def\Fix{{\rm Fix}}
\def\GL{{\rm GL}}
\def\Ad{{\rm Ad}}
\def\Aut{{\rm Aut}}
\def\der{{\rm der}}
\def\ad{{\rm ad}}
\def\diag{{\rm diag}}
\def\Mink{{\rm Mink}}
\def\Tay{{\rm Tay}}
\def\lin{{\rm lin}}
\def\switch{{\rm switch}}
\def\SOtwortwo{{\SO(2)\ltimes\R^2}}
\def\O{{\rm O}}
\def\OB{{\rm OB}}
\def\SO{{\rm SO}}
\def\SL{{\rm SL}}
\def\PSL{{\rm PSL}}
\def\SBF{{\rm SBF}}
\def\STF{{\rm STF}}
\def\SM{{\rm SM}}
\def\QF{{\rm QF}}
\def\Id{{\rm Id}}
\def\dom{{\rm dom}}
\def\im{{\rm im}}
\def\dd{{\rm dd}}
\def\dct{{\rm dct}}
\def\sym{{\rm sym}}
\def\Sym{{\rm Sym}}
\def\as{{\rm as}}
\def\mm{{\rm mm}}
\def\cof{{\rm cof}}
\def\adj{{\rm adj}}
\def\det{{\rm det}}
\def\DET{{\rm DET}}
\def\ADJ{{\rm ADJ}}
\def\DNZ{{\hbox{\rm DNZ}}}
\def\CVZ{{\hbox{\rm CVZ}}}
\def\BNZ{{\hbox{\rm BNZ}}}
\def\ev{{\rm ev}}
\def\sl{{\rm s\ell}}
\def\JO{{\rm JO}}
\def\TL{{\rm TL}}

%misc
\def\vep{{\varepsilon}}
\def\bksl{{\backslash}}
\def\sigmaalg{{\rm s.alg}}
\def\drta{\dashrightarrow}
\def\ltlt{{<\!\!<}}
\def\phx{\phantom{x}}

\def\Xint#1{\mathchoice
{\XXint\displaystyle\textstyle{#1}}%
{\XXint\textstyle\scriptstyle{#1}}%
{\XXint\scriptstyle\scriptscriptstyle{#1}}%
{\XXint\scriptscriptstyle\scriptscriptstyle{#1}}%
\!\int}
\def\XXint#1#2#3{{\setbox0=\hbox{$#1{#2#3}{\int}$ }
\vcenter{\hbox{$#2#3$ }}\kern-.6\wd0}}
\def\ddashint{\Xint=}
\def\dashint{\Xint-}

\begin{document}

\centerline{\bf Homework for MATH 4603 (Advanced Calculus I)}
\centerline{\bf Fall 2022}
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 13: Due on Tuesday 13 December}
\vskip.1in\noindent
13-1. Show: Let $S,T\subseteq\R$. \quad Then $\JO_{S\cup T}\,\,\le\,\,\JO_S\,+\,\JO_T$.
\vskip.1in\noindent
13-2. Show: $\forall k\in\N$,\\
\phx\qquad\qquad\qquad $\forall S_1,\ldots,S_k\subseteq\R$, \quad $\JO_{S_1\cup\cdots\cup S_k}\,\,\le\,\,\JO_{S_1}\,+\,\cdots\,+\,\JO_{S_k}$.
\vskip.1in\noindent
13-3. Show: Let $k\in\N$. \quad
Assume: $\,$ $\forall\alpha\in\R$, $\forall\beta\ge\alpha$, $\,$ $\JO_{[\alpha;\beta]}^k\ge\beta-\alpha$.\\
Let $a\in\R$. $\,$ Let $b\ge a$. $\,$ Let $I:=[a;b]$. $\,$ Then $\JO_I^{k+1}\ge b-a$.
\vskip.1in\noindent
13-4. Show: Let $T\subseteq\R$. \quad Then $\Int\,T$ is open.
\vskip.1in\noindent
13-5. Show: Let $S\subseteq\R$. \quad Then $\Cl\,S$ is closed.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 12: Due on Tuesday 6 December}
\vskip.1in\noindent
12-1. Using the PMI, show: Let $a\in\R$. Then:\\
\phx\qquad\qquad $\forall j\in\N$, \qquad $((\bullet)^j)^\T_a-j\cdot a^{j-1}\cdot(\bullet)\in\scrlito(1)$.
\vskip.1in\noindent
12-2. Show: \quad Let $j\in\N$. \quad Then $((\bullet)^j)'=j\cdot(\bullet)^{j-1}$.
\vskip.1in\noindent
12-3. Show: Let $f:\R\drta\R$, $y\in\R_0^\times$.\\
\phx\qquad\qquad\qquad Assume: \quad as $x\to-\infty$, $f_x\to y$.\\
\phx\qquad\qquad\qquad Then: \qquad as $x\to-\infty$, $(1/f)_x\to1/y$.
\vskip.1in\noindent
12-4. Show: Let $f:\R\drta\R$, $a\in\D_f$.\\
\phx\qquad\qquad\qquad Assume: \quad $f$ has a local strict-maximum at $a$.\\
\phx\qquad\qquad\qquad Then: \qquad $f_a^\T$ has a local strict-maximum at $0$.
\vskip.1in\noindent
12-5. Show: Let $s,t\in\R^\N$.\\
\phx\qquad\qquad\qquad Assume: \quad $\I_s$ and $\I_t$ are both bounded.\\
\phx\qquad\qquad\qquad Then: \qquad $\exists$strictly-increasing $\ell\in\N^\N$ s.t.\\
\phx\qquad\qquad\qquad\qquad\qquad\qquad\qquad $s\circ\ell$ and $t\circ\ell$ are both convergent.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 11: Due on Tuesday 29 November}
\vskip.1in\noindent
11-1. Show: Define $r:\R_0^\times\to\R$ by: $\forall x\in\R_0^\times$, $\,$ $r_x=1/x$.\\
Then: $\forall a\in\R_0^\times$, $\,$ $\ds{r'_a=-\frac{1}{a^2}}$.
\vskip.1in\noindent
11-2. Show: Let $g:\R\drta\R$, $a\in\D'_g$.\\
Assume $g_a\ne0$. \quad Then: $\ds{\left(\frac{1}{g}\right)'_a=-\frac{g'_a}{g_a^2}}$.
\vskip.1in\noindent
11-3. Show: Let $f,g:\R\drta\R$, $a\in\D'_f\cap\D'_g$.\\
Assume $g_a\ne0$. \quad Then: $\ds{\left(\frac{f}{g}\right)'_a=\frac{g_a\cdot f'_a-f_a\cdot g'_a}{g_a^2}}$.
\vskip.1in\noindent
11-4. Show: Let $U,V\subseteq\R$ both be open. Then:\\
\phx\quad (i) $U\cup V$ is open \qquad\qquad and \qquad\qquad (ii) $U\cap V$ is open.
\vskip.1in\noindent
11-5. Show: Let $f:\R\drta\R$. \quad
Assume: $\forall$open $U\subseteq\R$, $\,$ $f^*U$ is open.\\
Then: $f$ is continuous.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 10: Due on Tuesday 22 November}
\vskip.1in\noindent
10-1. Show: Let $w,x\in\R^\N$, $\ell\in\N^\N$, $q\in\R$.\\
Assume: $($ $\ell$ is strictly-increasing $)$ $\&$ $(\,w\circ\ell\to q\,)$.\\
Assume: $\forall j\in\N$, $|w_j-x_j|<1/j$.\\
Then: $x\circ\ell\to q$.
\vskip.1in\noindent
10-2. Show: Let $f,g:\R\drta\R$.\\
Assume: as $x\to\infty$, $f_x\to-\infty$.\\
Assume: as $x\to\infty$, $g_x\to-\infty$.\\
Then: as $x\to\infty$, $(f\cdot g)_x\to\infty$.
\vskip.1in\noindent
10-3. Show: Let $f:=(\bullet)^3$. Then $D_2f=3\cdot2^2\cdot(\bullet)$.
\vskip.1in\noindent
10-4. Show: Let $f:=(\bullet)^3$. Then $\forall x\in\R$, $f'_x=3x^2$.
\vskip.1in\noindent
10-5. Show: Let $f:\R\drta\R$, $a,c\in\R$.\\
Then $D_a(c\cdot f)\,\,\,\,=^*\,\,c\cdot(D_af)$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 9: Due on Tuesday 15 November}
\vskip.1in\noindent
9-1. Show: Define $f:\R\drta\R$ by: \quad $\forall x\in\R$, $\,$ $f_x\,\,=\,\,1/x$.\\
Then: \quad as $x\to-\infty$, $\,$ $f_x\to0$.
\vskip.1in\noindent
9-2. Show: Let $D\subseteq\R$, $s\in(\R^D)^\N$, $f\in\R^D$.\\
Assume: \quad $\forall j\in\N$, $s_j$ is continuous.\\
Assume: \quad $s\to f$ uniformly.\\
Then: \qquad $f$ is continuous.
\vskip.1in\noindent
9-3. Show: Let $f,g:\R\drta\R$, $k\in\N_0$.\\
Assume: \quad $(\,\,f\in\scrlito(k)\,\,)\,\,\&\,\,(\,\,\hbox{near }0,\,\,f=g\,\,)$.\\
Then: \qquad\quad $g\in\scrlito(k)$.
\vskip.1in\noindent
9-4. Show: Let $s,t\in\R^\N$.\\
Assume: \quad $(\,\,s\to\infty\,\,)\,\,\&\,\,(\,\,\forall j\in\N,\,\,t_j\ge s_j\,\,)$.\\
Then: \qquad\quad $t\to\infty$.
\vskip.1in\noindent
9-5. Show:\\
Let $a\,:=\,(\,1\,,\,1/2\,,\,1/2\,,\,1/4\,,\,1/4\,,\,1/4\,,\,1/4\,,\,1/8\,,\,\ldots)$.\\
\phx NOTE: The terms of $a$ are, in order:\\
\phx\qquad\qquad\qquad a single ``$1$'', \quad then two ``$1/2$''s,\\
\phx\qquad\qquad\qquad then four ``$1/4$''s, \quad then eight ``$1/8$''s,\\
\phx\qquad\qquad\qquad then sixteen ``$1/16$''s, \quad then thirty-two ``$1/32$''s, \quad {\it etc.}\\
Let $b\,:=\,(\,1\,,\,1/2\,,\,1/3\,,\,1/4\,,\,1/5\,,\,1/6\,,\,1/7\,,\,1/8\,,\,\ldots)$.\\
Define $s,t\in\R^\N$ by: $\forall j\in\N$,\\
\phx\qquad $s_j=a_1+\cdots+a_j$ \quad and \quad $t_j=b_1+\cdots+b_j$.\\
Then: $\,\,\,$(i) $\I_s$ is unbounded \qquad\quad and\\
\phx\qquad $\,\,\,\,\,$(ii) $\forall j\in\N$, $\,$ $t_j>s_j/2$ \qquad and\\
\phx\qquad $\,\,\,\,$(iii) $t\to\infty$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 8: Due on Wednesday 9 November}
\vskip.1in\noindent
8-1. Show: Let $f\in\scrlito(3)$, $g\in\scrlito(4)$. \quad
Then: $f\cdot g\in\scrlito(7)$.
\vskip.1in\noindent
8-2. Show: Let $s\in\R^\N$.\\
Assume $\I_s$ is bounded. \quad Then $s$ is subconvergent.
\vskip.1in\noindent
8-3. Show: Let $s\in\R^\N$.\\
Assume $s$ is convergent. \quad Then $s$ is Cauchy.
\vskip.1in\noindent
8-4. Show: Let $s\in\R^\N$.\\
Assume $s$ is Cauchy and subconvergent. \quad Then $s$ is convergent.
\vskip.1in\noindent
8-5. Show: Let $s\in\R^\N$.\\
Assume $s$ is Cauchy. \quad Then $s$ is convergent.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 7: Due on Tuesday 1 November}
\vskip.1in\noindent
7-1. Show: Let $k\in\N^\N$. \quad Assume $k$ is strictly-increasing.\\
Then: \quad $\forall j\in\N$, $\,$ $k_j\ge j$.
\vskip.1in\noindent
7-2. Show: Let $s,t\in\R$. \quad Assume $s<t$.\\
Then: \quad $\exists x\in\Q$ s.t.~$s<x<t$.
\vskip.1in\noindent
7-3. Show: Let $s,t,u\in\R^\N$, $a\in\R$.\\
Assume: $\forall j\in\N$, $s_j\le t_j\le u_j$.\\
Assume: $s\to a$ and $u\to a$.\\
Then: $t\to a$.
\vskip.1in\noindent
7-4. Show: Let $f,g:\R\drta\R$, $a\in\R$.\\
Assume: as $x\to a$, $f_x\to-\infty$.\\
Assume: as $x\to a$, $g_x\to-\infty$.\\
Then: as $x\to a$, $(f\cdot g)_x\to\infty$.
\vskip.1in\noindent
7-5. Show: Let $f,g:\R\drta\R$, $q\in\D_{f\cdot g}$.\\
Then: $(f\cdot g)_q^\T\,\,=\,\,f_q^\T\cdot g_q\,\,+\,\,f_q\cdot g_q^\T\,\,+\,\,f_q^\T\cdot g_q^\T$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 6: Due on Tuesday 25 October}
\vskip.1in\noindent
6-1. Show: $\forall a\in\R$, $\exists x\in\R$ s.t.~$x^5+x^3+x=a$.
\vskip.1in\noindent
6-2. Show: Let $f,g:\R\drta\R$, $a,b\in\R$.\\
Assume: as $x\to a$, $f_x\to b$. \qquad
Assume: $g$ is continuous at $b$.\\
\phx\qquad\qquad Then: as $x\to a$, $(g\circ f)_x\to g_b$.
\vskip.1in\noindent
6-3. Show: Let $f:\R\drta\R$, $a,b\in\D_f$.\\
Assume $a\ne b$. \qquad Let $m:=\DQ_f(a,b)$.\\
Define $g:\R\drta\R$ by: \qquad $\forall x\in\R$, \quad $g_x=f_x-mx$.\\
Then: $g_a=g_b$.
\vskip.1in\noindent
6-4. Show: Let $f:\R\drta\R$, $a\in\D_f$.\\
Assume $f_a^\T$ is continuous at $0$. \quad Then $f$ is continuous at $a$.
\vskip.1in\noindent
6-5. Show: Let $f,g:\R\drta\R$, $a\in\D_{g\circ f}$.
Then $(g\circ f)_a^\T\,=\,(g_{f_a}^\T)\,\circ\,(f_a^\T)$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 5: Due on Wednesday 19 October}
\vskip.1in\noindent
5-1. Show: Let $f,g:\R\drta\R$, $a\in\R$.\\
Assume $f$ and $g$ are both continuous at $a$.\\
Then $f+g$ is continuous at $a$.
\vskip.1in\noindent
5-2. Show: Let $f,g:\R\drta\R$, $a\in\R$.\\
Assume \quad $f$ is continuous at $a$ \quad and \quad $g$ is continuous at $f_a$.\\
Then $g\circ f$ is continuous at $a$.
\vskip.1in\noindent
5-3. Show: Let $f:\R\drta\R$, $s\in(\D_f)^\N$, $a\in\R$.\\
Assume \quad $f$ is continuous at $a$ \quad and \quad $s\to a$. \qquad Then $f\circ s\to f_a$.
\vskip.1in\noindent
5-4. Show: let $f:\R\drta\R$.\\
Assume $f$ is Lipschitz. \quad Then $f$ is uniformly continuous.
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5-5. Show: let $f:\R\drta\R$.\\
Assume $f$ is uniformly continuous. \quad Then $f$ is continuous.
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\centerline{Homework 4: Due on Tuesday 11 October}
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4-1. Show: $(1,2,3,\ldots)\to\infty$.
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4-2. Show: Let $s\in\R^\N$, $c<0$.\\
Assume $s\to\infty$. \qquad Then $c\cdot s\to-\infty$.
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4-3. Show: Let $s,t\in\R^\N$.\\
Assume $s\to\infty$ and $t\to\infty$. \qquad Then $s+t\to\infty$.
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4-4. Show: Let $s\in(\R_0^\times)^\N$.\\
Assume $s\to\infty$. \qquad Then $1/s\to0$.
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4-5. Show: Let $s\in(\R_0^\times)^\N$, $c\in\R$.\\
Assume $s\to-\infty$. \qquad Then $c/s\to0$.
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\centerline{Homework 3: Due on Tuesday 4 October}
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3-1. Using the $0$-PMI, show: Let $x\in\R$. Then: $\forall j\in\N_0$, $|x^j|=|x|^j$.
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3-2. Show: $\forall\varepsilon>0$, $\exists\delta>0$ s.t., $\forall x\in\R$,\\
\phx\qquad\qquad\qquad\qquad $(\,\,|x-4|<\delta\,\,)\,\,\Rightarrow\,\,(\,\,|x^3-5x^2+4x|<\varepsilon\,\,)$.
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3-3. Show: Define $f:\R\to\R$ by: $\forall x\in\R$, $f_x=x^2$.\\
\phx\quad $\,$ Then: $\forall a\in\R$, $\forall\varepsilon>0$, $\exists\delta>0$ s.t., $\forall x\in\R$,\\
\phx\qquad\qquad\qquad\qquad\qquad $(\,\,|x-a|<\delta\,\,)\,\,\Rightarrow\,\,(\,\,|f_x-f_a|<\varepsilon\,\,)$.
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3-4. Show: $\forall M\in\R$, $\exists K\in\N$ s.t., $\forall j\in\N$,\\
\phx\quad\qquad\qquad\qquad\qquad $(\,\,j\ge K\,\,)\,\,\Rightarrow\,\,(\,\,j^4-7j-9>M\,\,)$.
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3-5. Show: \qquad Let $x,y\in\R$. \quad Then: $\,\,$ $|\,\,|y|\,\,-\,\,|x|\,\,|\,\,\,\,\le\,\,\,\,|\,\,y\,\,-\,\,x\,\,|$.
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\centerline{Homework 2: Due on Tuesday 27 September}
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2-1. Show: Let $S\subseteq\R^*$. Then: $\sup_S\,\,\,\,=^*\,\,\max_S$.
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2-2. Show: $\forall M\in\R$, $\exists N\in\R$ s.t., $\forall x\in\R$,\\
\phx\qquad\qquad\qquad $(\,\,x<N\,\,)\,\,\Rightarrow\,\,(\,\,x^2>M\,\,)$.
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2-3. Show: $\forall M\in\R$, $\exists K\in\N_0$ s.t., $\forall j\in\N_0$,\\
\phx\qquad\qquad\qquad $(\,\,j\ge K\,\,)\,\,\Rightarrow\,\,(\,\,2^j>M\,\,)$.
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2-4. Show: $\forall\varepsilon>0$, $\exists K\in\N_0$ s.t., $\forall j\in\N_0$,\\
\phx\qquad\qquad\qquad $\ds{(\,\,j\ge K\,\,)\,\,\Rightarrow\,\,\left(\,\,\frac{1}{2^j}<\varepsilon\,\,\right)}$.
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2-5. Using the $0$-PMI, show: $\forall k\in\N_0$,\\
\phx\qquad\qquad\qquad $1+2+4+8+\cdots+2^k\,\,=\,\,2^{k+1}-1$.
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\centerline{Homework 1: Due on Tuesday 20 September}
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1-1. Show: $\forall\varepsilon>0$, $\exists\delta>0$ s.t.~$3\delta^8+7\delta^6+2\delta^4\le6\varepsilon$.
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1-2. Show: Let $\varepsilon>0$. Then $\exists\delta>0$ s.t.~$3\delta^8+7\delta^6+2\delta^4\le6\varepsilon$.
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1-3. Show: $\forall N\in\R$, $\exists\delta>0$ s.t.~$\forall x\in\R$,\\
\phx\qquad\qquad $(\,-\delta<x<0\,)\,\Rightarrow\,(\,1/x<N\,)$.
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1-4. Show: Let $N\in\R$. Then $\exists\delta>0$ s.t.~$\forall x\in\R$,\\
\phx\qquad\qquad $(\,-\delta<x<0\,)\,\Rightarrow\,(\,1/x<N\,)$.
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1-5. Using the PMI, show:\\
\phx\qquad\qquad $\ds{\forall k\in\N,\qquad1^2+2^2+\cdots+k^2=\frac{k(k+1)(2k+1)}{6}}$.
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