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\def\P{{\mathbb P}}

% BOLD
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% 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}}
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    {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}}%\scriptscriptstyle
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%operators
\def\opstar{\mathop{*}}
\def\LIMS{{\rm LIMS}}
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\def\SM{{\rm SM}}
\def\fincap{{\hbox{{\rm fin}}\,\cap}}
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\def\diam{{\rm diam}}
\def\linf{{\rm liminf\,}}
\def\lsup{{\rm limsup\,}}
\def\acc{{\rm acc\,}}
\def\dom{{\rm dom\,}}
\def\im{{\rm im\,}}
\def\barR{{\overline{\R}}}
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\def\Sym{{\rm Sym}}
\def\as{{\rm as}}
\def\id{{\rm id}}
\def\dlin{{\rm dlin}}
\def\dct{{\rm dct}}
\def\SBF{\hbox{SBF}}
\def\Int{{\rm Int}}
\def\IntD{{\rm IntD}}
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\begin{document}
\centerline{\bf Homework for MATH 4604 (Advanced Calculus II)}
\centerline{\bf Spring 2018}
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 14: Due on Tuesday 1 May}
\vskip.1in\noindent
14-1. Let $W$ be a normed vector space, let $f:\R\drta W$ and let $p\in\R$.
Show: $(f_p^T)'(0)=f'(p)$.
\vskip.1in\noindent
14-2. Let $V$, $W$ be finite dimensional vector spaces,
$\alpha,\beta:V\drta W$, $p\ge0$.
Assume: $\alpha=\beta$ near $0_V$ and $\alpha\in\lito_p(V,W)$.
Show: $\beta\in\lito_p(V,W)$.
\vskip.1in\noindent
14-3. Let $\delta>0$, let $I:=(-\delta,\delta)$ and let $\alpha:\R\drta\R$.
Assume: $\forall x\in I$, $\alpha(x)\in[0|x]$.
Show that $\alpha\in\bigo_1(\R,\R)$.
\vskip.1in\noindent
14-4. Let $k\in\N_0$ and let $f\in\lito_k(\R,\R)$.
Let $g:\R\drta\R$.
Assume that $g'=f$ near $0$.
Assume that $g(0)=0$.
Show that $g\in\lito_{k+1}(\R,\R)$.
\vskip.1in\noindent
14-5. Let $V$,  $W$ be finite dimensional vector spaces.
Let $f,g:V\drta W$.
Let $p\in V$.
Assume: $f=g$ near $p$.
Show: $\LINS_pf\subseteq\LINS_pg$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 13: Due on Tuesday 24 April}
\vskip.1in\noindent
13-1. Let $V$, $W$ be finite dimensional VSs.
Show: $L(V,W)\subseteq\bigo_1(V,W)$.
\vskip.1in\noindent
13-2. Let $m,n\in\N$.
Let $V:=(\R^m,|\bullet|_{m,1})$ and let $W:=(\R^n,|\bullet|_{n,1})$.
Let $Z$ be a normed vector space.
Let $*\in B(V,W,Z)$.
Show: $\exists K\ge0$ such that, $\forall v\in V$, $\forall w\in W$,
\quad $|v*w|_Z\,\le\,K\cdot|v|_V\cdot|w|_W$.
\vskip.1in\noindent
13-3. Let $\ell,m,n\in\N$.
Let $U:=(\R^\ell,|\bullet|_{\ell,1})$, let $V:=(\R^m,|\bullet|_{m,1})$ and let $W:=(\R^n,|\bullet|_{n,1})$.
Let $Z$ be a normed vector space.
Let $F\in T(U,V,W,Z)$.
Show: $\exists K\ge0$ such that, $\forall u\in U$, $\forall v\in V$, $\forall w\in W$,
$$|F(u,v,w)|_Z\quad\le\quad K\cdot|u|_U\cdot|v|_V\cdot|w|_W.$$
\vskip.1in\noindent
13-4. Let $U$, $V$ and $W$ be finite dimensional normed vector spaces.
Let $Z$ be a normed vector space.
Let $F\in T(U,V,W,Z)$.
Show: $\exists K\ge0$ such that, $\forall u\in U$, $\forall v\in V$, $\forall w\in W$,
$$|F(u,v,w)|_Z\quad\le\quad K\cdot|u|_U\cdot|v|_V\cdot|w|_W.$$
\vskip.1in\noindent
13-5. Let $S$ $V$, $W$ and $Z$ all be finite dimensional vector spaces,
and let $*\in B(V,W,Z)$.
Show: $\ds{[\bigo(S,V)]\,\opstar_S\,[\bigo(S,W)]\,\,\subseteq\,\,\bigo(S,Z)}$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 12: Due on Tuesday 17 April}
\vskip.1in\noindent
12-1. Let $V$ and $W$ both be finite dimensional vector spaces,
and let $|\bullet|,\|\bullet\|\in\scrn(W)$.
Show: $\bigo(V,W,|\bullet|)\subseteq\bigo(V,W,\|\bullet\|)$.
\vskip.1in\noindent
12-2. Let $V$ and $W$ both be finite dimensional vector spaces,
and let $\alpha:V\dashrightarrow W$.
Assume that $\dom[\alpha]\in\scrn_V(0_V)$.
Assume that $\alpha$ is continuous at $0_V$.
Show: $\alpha\in\bigo(V,W)$.
\vskip.1in\noindent
12-3. Let $V$ and $W$ both be finite dimensional vector spaces,
and let $|\bullet|,\|\bullet\|\in\scrn(V)$, and let $p>0$.
Show:
\begin{itemize}
\item[(1)]$\lito_p(V,W,|\bullet|)\subseteq\lito_p(V,W,\|\bullet\|)$ \qquad and
\item[(2)]$\bigo_p(V,W,|\bullet|)\subseteq\bigo_p(V,W,\|\bullet\|)$.
\end{itemize}
\vskip.1in\noindent
12-4. Let $V$ and $W$ both be finite dimensional vector spaces,
and let $p,q\ge0$.
Assume $p<q$.
Show: $\lito_p(V,W)\supseteq\bigo_q(V,W)$.
\vskip.1in\noindent
12-5. Let $p,q\ge0$.
Show:
\begin{itemize}
\item[](\,\,[\,\,$(\lito_p(\R,\R))\cdot(\lito_q(\R,\R))\subseteq\lito_{p+q}(\R,\R)$\,\,]\qquad and\\
\phantom{x}\,[\,\,$(\lito_q(\R,\R))\circ(\lito_p(\R,\R))\,\subseteq\,\lito_{qp}(\R,\R)$\,\,]\,\,).
\end{itemize}
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 11: Due on Tuesday 10 April}
\vskip.1in\noindent
11-1. Let $S\subseteq\R$.
Assume that $S$ has a minimum.
(That is, assume: $\exists a\in S$ s.t.~$a\le S$.)
Show that $\inf S=\min S\in S$.
\vskip.1in\noindent
11-2. Let $U$ be a vector space,
let $|\bullet|,\|\bullet\|\in\scrn(V)$
and let $S\subseteq U$.
Let $V:=(U,|\bullet|)$ and let $W:=(U,\|\bullet\|)$.
Assume both that $|\bullet|<<\|\bullet\|$,
and that $S$ is bounded in $W$.
Show that $S$ is bounded in $V$.
\vskip.1in\noindent
11-3. Let $V$ and $W$ be normed vector spaces,
and let $T\in L(V,W)$.
Assume that $T:V\to W$ is bounded below.
Show that $T$ is 1-1.
\vskip.1in\noindent
11-4. Let $T\in L(\R^4,\R^4)$ be defined by $Tx=(6x_1,5x_2,8x_3,7x_4)$,
and define $V:=(\R^4,|\bullet|_{4,2})$.
Show that $\ckT_{VV}=5$
and that $\hatT_{VV}=8$.
\vskip.1in\noindent
11-5. Let $V$ and $W$ be normed vector spaces
and let $T:V\hookrightarrow>W$ be a vector space isomorphism.
Assume $T:V\to W$ is bounded below.
Show that $T^{-1}:W\to V$ is bounded.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 10: Due on Tuesday 3 April}
\vskip.1in\noindent
10-1. Let $V$ and $W$ be normed vector spaces,
$T\in L(V,W)$ and $\varepsilon>0$.
Show:
$[\,T\hbox{ is }\varepsilon\hbox{-bounded below}\,]\,\Leftrightarrow\,
[\,\forall x\in V,\,|Tx|_W\ge\varepsilon|x|_V\,]$.
\vskip.1in\noindent
10-2. Let $m\in\N$.
Show: $|\bullet|_{m,\infty}\le|\bullet|_{m,2}\le|\bullet|_{m,1}\le m\cdot|\bullet|_{m,\infty}$.
\vskip.1in\noindent
10-3. Let $V$ be a normed vector space, let $Z$ be a topological space, let $\mu:V\drta Z$ and let $p\in V$.
Let $\lambda:=\mu(p+\bullet)$.
Show: $\ds{\lim_{0_V}\,\lambda=\lim_p\,\mu}$.
\vskip.1in\noindent
10-4. Let $X$ be a metric space, let $D\subseteq X$ and let $p\in\Int_XD$.
Show that there exists $B\in\scrb_X(p)$ such that $B\subseteq D$.
\vskip.1in\noindent
10-5. Let $g:\R\drta\R$.
Let $p,q\in\R$.
Assume $g$ has a local unique min at $p$ in $\R$.
Show: $g+C_\R^q$ has a local unique min at $p$ in~$\R$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 9: Due on Tuesday 27 March}
\vskip.1in\noindent
9-1. Let $V$ and $W$ be normed vector spaces, $T\in L(V,W)$ and $K\ge0$.
Show: \qquad
$[\,\,T\hbox{ is }K\hbox{-bounded}\,\,]\,\,\,\,\Leftrightarrow\,\,\,\,[\,\,\forall x\in V,\,\,\,|Tx|_W\,\le\,K\cdot|x|_V\,\,]$.
\vskip.1in\noindent
9-2. Let $V$ and $W$ be normed vector spaces and let $K\ge0$.
Let $T\in L(V,W)$ be $K$-bounded.
Show: $T$ is $K$-Lipschitz.
That is, show:
$$\forall x,y\in V,\qquad\qquad|\,\,[T(x)]\,\,-\,\,[T(y)]\,\,|_W\,\,\le\,\,K\cdot|\,x\,-\,y\,|_V.$$
\vskip.1in\noindent
9-3. Let $Y$ and $Z$ be metric spaces and let $K>0$.
Let $f:Y\dashrightarrow Z$ be a $K$-Lipschitz function.
Let $x\in\dom[f]$ and let $r>0$.
Show that: $f_*(B_Y(x,r))\subseteq B_Z(f(x),Kr)$.
\vskip.1in\noindent
9-4. Let $V$ and $W$ be normed vector spaces and let $T\in L(V,W)$.
Assume that $T$ is continuous at $0_V$.
Show that $T$ is bounded.
\vskip.1in\noindent
9-5. Let $f,g:\R\dashrightarrow\R$.
Assume: $\exists M\in\R$ s.t.~$(M,\infty)\subseteq\dom[g'/f'\,]$.
Assume: $\ds{\lim_\infty\,f=0=\lim_\infty\,g}$.
Show: $\ds{\lim_\infty\,(g/f)\,\,=^*\,\,\lim_\infty\,(g'/f')}$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 8: Due on Tuesday 20 March}
\vskip.1in\noindent
8-1. Let $W$ be a vector space and let $u,v,v'\in W$.
Assume that $\R v=\R v'$ and that $u\|v$.
Show that $u\|v'$.
\vskip.1in\noindent
8-2. Let $u,v\in\R^2$.
Show:
$$[\,\,u\|v\,\,]\quad\Leftrightarrow\quad[\,\,(u=0_2)\hbox{ or }(v=0_2)\hbox{ or }(\slp\,u=\slp\,v)\,\,].$$
\vskip.1in\noindent
8-3. Let $u,v\in\R^2$.
Show:
$$(\,\,u\|v\,\,)\quad\Leftrightarrow\quad
\left(\,\,\Det\left[\begin{array}{c}u\\v\end{array}\right]=0\,\,\right).$$
\vskip.1in\noindent
8-4. Let $V$ and $W$ be normed vector spaces.
Let $x:\R\dashrightarrow V$ and $y:\R\dashrightarrow W$.
Let $p\in\R$.
Assume that $x$ and $y$ are both continuous at~$p$.
Show that $(x,y)$ is continuous at $p$.
\vskip.1in\noindent
8-5. Let $V$ and $W$ be normed vector spaces.
Let $x:\R\dashrightarrow V$ and $y:\R\dashrightarrow W$.
Let $p\in\LPD_\R(x,y)$.
Show: $(x,y)'(p)\,\,=^*\,\,(\,x'(p)\,,\,y'(p)\,)$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 7: Due on Tuesday 6 March}
\vskip.1in\noindent
7-1. Let $f:\R\dashrightarrow\R$, $p\in\R$ and $\delta>0$.
Assume
\begin{itemize}
\item[(1)]$(p-\delta,p+\delta)\,\,\subseteq\,\,\dom[f]$,
\item[(2)]$f$ is strictly decreasing on $(p-\delta,p]$ \qquad\qquad and
\item[(3)]$f$ is strictly increasing on $[p,p+\delta)$.
\end{itemize}
Show that $f$ has a local unique minimum at $p$ in $\R$.
\vskip.1in\noindent
7-2. Let $g:\R\dashrightarrow\R$ and $p\in\IntD_\R g$.
Assume both that $g(p)=0$ and that $g'(p)>0$.
Show: $\exists\delta>0$ such that
all three of the following hold:
\begin{itemize}
\item[(A)]$(p-\delta,p+\delta)\,\,\subseteq\,\,\dom[g]$,
\item[(B)]$g<0$ on $(p-\delta,p)$ \qquad\qquad and
\item[(C)]$g>0$ on $(p,p+\delta)$.
\end{itemize}
\vskip.1in\noindent
7-3. Let $f:\R\dashrightarrow\R$ and let $p\in\IntD_\R(f')$.
Assume both that $f'(p)=0$ and that $f''(p)>0$.
Show: $f$ has a local unique minimum at $p$ in~$\R$.
\vskip.1in\noindent
7-4. Let $f:\R\dashrightarrow\R$ and let $p\in\IntD_\R(f')$.
Assume both that $f'(p)=0$ and that $f''(p)<0$.
Show: $f$ has a local unique maximum at $p$ in~$\R$.
\vskip.1in\noindent
7-5. Let $f:\R\dashrightarrow\R$.
Asume both that $0\in\IntD_\R f$ and that $f(0)=0$.
Define $g:\R\to\R$ by $g(x)=x^4$.
Assume that $f/g\to1$ near $0$.
Show that $f$ has a local unique minimum at $0$ in $\R$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 6: Due on Tuesday 27 February}
\vskip.1in\noindent
6-1. Let $f:\R\dashrightarrow\R$ be continuous, and let $I\subseteq\dom[f]$.
Assume that $I$ is an interval.
Show: $f_*(I)$ is an interval.
\vskip.1in\noindent
6-2. Let $X,Y\subseteq\R$.
Let $f:X\hookrightarrow>Y$ be continuous.
Let $X_0:=\Int_\R X$.
Show that $f^{-1}$ is continuous on $f_*(X_0)$.
\vskip.1in\noindent
6-3. Let $f:\R\dashrightarrow\R$ be str.~increasing.
Show: $f^{-1}$ is str.~increasing.
\vskip.1in\noindent
6-4. Let $X,Y\subseteq\R$.
Let $f:X\hookrightarrow>Y$ be strictly increasing.
Let $a\in\R$.
Assume: $X=[a,\infty)$.
Show: $f^{-1}$ is continuous at $f(a)$.
\vskip.1in\noindent
6-5. Let $X,Y\subseteq\R$.
Let $f:X\hookrightarrow>Y$ be strictly increasing.
Let $a\in\R$.
Assume: $\exists b\in(a,\infty)$ s.t.~$X=[a,b)$.
Show: $f^{-1}$ is continuous at $f(a)$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 5: Due on Tuesday 20 February}
\vskip.1in\noindent
5-1. Show: $\forall w,x\in\R$,\quad
$\cos(w+x)\,\,=\,\,(\cos w)\cdot(\cos x)\,-\,(\sin w)\cdot(\sin x)$.
\vskip.1in\noindent
5-2. Show: $\forall x\in\R$,
\begin{eqnarray*}
\sin(2x)&=&2\cdot(\sin x)\cdot(\cos x)\qquad\qquad\hbox{and}\\
\cos(2x)&=&(\cos^2 x)\,-\,(\sin^2x).
\end{eqnarray*}
\vskip.1in\noindent
5-3. Let $f:\R\to\R$, $a:=f(0)$.
Assume $f'=f$.
Show: $f=a\cdot\exp$.
\vskip.1in\noindent
5-4. Show:
$\forall w,x\in\R$, \quad
$\exp(w+x)\,\,=\,\,(\exp w)\cdot(\exp x)$.
\vskip.1in\noindent
5-5. Let $f:\R\to\R$.
Assume that $f'=f$.
Let $g:=f^2$.
Show: $g'=2g$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 4: Due on Tuesday 13 February}
\vskip.1in\noindent
4-1. Let $m\in\R$.
Define $\lambda:\R\to\R$ by $\lambda(x)=mx$.
Show: $\lambda'=C_\R^m$.
\vskip.1in\noindent
4-2. Let $S\subseteq\R$.
Show: $-(\LP_\R S)=\LP_\R(-S)$.
\vskip.1in\noindent
4-3. Let $f:\R\dashrightarrow\R$, $p\in\R$.
Define $f_0:\R\dashrightarrow\R$ by $f_0(x)=f(-x)$.
Let $p_0:=-p$.
Assume that $f$ has a local maximum at $p$ in $\R$.
Show that $f_0$ has a local maximum at $p_0$ in $\R$.
\vskip.1in\noindent
4-4. Let $f:\R\dashrightarrow\R$, $p\in\R$.
Define $f_1:\R\dashrightarrow\R$ by $f_1(x)=-[f(x)]$.
Assume that $f$ has a local minimum at $p$ in $\R$.
Show that $f_1$ has a local maximum at $p$ in $\R$.
\vskip.1in\noindent
4-5. Let $f:\R\dashrightarrow\R$, $p\in\dom[f']$.
Assume that $f$ has a local extremum at $p$.
Show that $f'(p)=0$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 3: Due on Tuesday 6 February}
\vskip.1in\noindent
3-1. Let $X$ be a topological space, let $S\subseteq X$ and let $p\in X$.
Show:
$$[\,\,p\in\Int_XS\,\,]\quad\Leftrightarrow\quad[\,\,S\in\scrn_X(p)\,\,].$$
\vskip.1in\noindent
3-2. Let $\phi:\R\dashrightarrow\R$, let $p\in\R$ and let $q>0$.
Assume: $\phi\to q$ near $p$.
Show: $\exists U\in\scrn_\R^\times(p)$ s.t.~$\phi_*(U)>0$.
\vskip.1in\noindent
3-3. Let $\alpha,\beta,\gamma:\R\dashrightarrow\R$.
Let $S\subseteq\dom[(\alpha,\beta,\gamma)]$.
Let $p\in\Int_\R S$ and let $q\in\R$.
Assume that $\alpha\le\beta\le\gamma$ on $S$.
Assume that $\alpha(p)=\gamma(p)$.
Assume that $\alpha'(p)=q=\gamma'(p)$.
Show: $\beta'(p)=q$.
\vskip.1in\noindent
3-4. Let $W$ be a normed vector space.
Let $f,g:\R\dashrightarrow W$.
Let $U$ be an open subset of $\R$.
Assume: $U\subseteq\dom[(f,g)]$.
Assume: $f=g$ on $U$.
Show: $f'=g'$ on $U$.
\vskip.1in\noindent
3-5. Define $f:\R\dashrightarrow\R$ by
$f(x)=[x^2][\sin(x^{-3})]$.
Let $\beta:=\adj_0^0f$.
Show: $\beta'(0)=0$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 2: Due on Tuesday 30 January}
\vskip.1in\noindent
2-1. Let $X$ and $Y$ be topological spaces.
Let $X_0\subseteq X$ and let $Y_0\subseteq Y$.
Let $f:X_0\dashrightarrow Y_0$.
Show:
$$[\,\,f\hbox{ is }(X,Y)\hbox{-continuous}\,\,]\quad\Rightarrow\quad[\,\,f\hbox{ is }(X_0,Y_0)\hbox{-continuous}\,\,].$$
\vskip.1in\noindent
2-2. Let $X$, $Y$ and $Z$ be topological spaces.
Let $f:X\dashrightarrow Y$ and let $g:X\dashrightarrow Z$.
Let $a\in X$, $b\in Y$ and $c\in Z$.
Assume:
\begin{itemize}
\item[(1)]$f\to b$ in $Y$ near $a$ in $X$ \qquad and
\item[(2)]$g\to c$ in $Z$ near $a$ in $X$.
\end{itemize}
Show: $(f,g)\to(b,c)$ in $Y\times Z$ near $a$ in $X$.
\vskip.1in\noindent
2-3. Show: $\forall z\in\R$, $\exists!x\in\R$ s.t.~$x^5+x^3=z$.
\vskip.1in\noindent
2-4. Let $X$ and $Y$ be topological spaces.
Let $f:X\dashrightarrow Y$.
Let $X_0\subseteq\dom[f]$.
Assume that $f$ is $(X,Y)$-continuous on $X_0$.
Show that $f|X_0$ is $(X_0,Y)$-continuous.
\vskip.1in\noindent
2-5. Let $f:\R\dashrightarrow\R$.
Let $\alpha,\beta\in\dom[f]$.
Assume that $f$ is semimonotone on $[\alpha|\beta]$.
Show: $f_*([\alpha|\beta])\subseteq[f(\alpha)|f(\beta)]$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 1: Due on Tuesday 23 January}
\vskip.1in\noindent
1-1. Let $X$ and $Y$ be metric spaces, and let $f:X\to Y$.
Assume that $f$ is not uniformly continuous.
Show: $\exists\varepsilon>0$, $\exists p,q\in X^\N$ s.t., $\forall j\in\N$,
$$[\,\,d_X\,(\,p_j\,,\,q_j\,)\,\,<\,\,1/j\,\,]\quad\hbox{and}\quad[\,\,d_Y\,(\,f(p_j)\,,\,f(q_j)\,)\,\,\ge\,\,\varepsilon\,\,].$$
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1-2. Let $K$ be a sequentially compact metric space,
let $Y$ be a metric space, and let $f:K\to Y$.
Assume that $f$ is not uniformly continuous.
Show: $\exists\varepsilon>0$, $\exists s,t\in K^\N$, $\exists u\in K$ s.t.
\begin{eqnarray*}
&&(\qquad[\,\,s_\bullet\to u\hbox{ in }K\,\,]\quad\hbox{and}\quad[\,\,t_\bullet\to u\hbox{ in }K\,\,]\quad\hbox{and}\\
&&\qquad\,\,\,[\,\,\forall j\in\N,\qquad\,d_Y\,(\,f(s_j)\,,\,f(t_j)\,)\,\,\ge\,\,\varepsilon\,\,]\qquad).
\end{eqnarray*}
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1-3. Let $X$ be a metric space, and let $s\in X^\N$.
Assume that $s_\bullet$ is convergent in $X$.
Show that $s_\bullet$ is Cauchy in $X$.
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1-4. Let $X$ be a metric space, and let $s\in X^\N$.
Assume that $s_\bullet$ is Cauchy and subconvergent in $X$.
Show that $s_\bullet$ is convergent in $X$.
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1-5. Let $X$ be a metric space, and let $s\in X^\N$.
Assume that $s_\bullet$ is Cauchy in $X$.
Show that $s_\bullet$ is bounded in $X$.
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\hrule
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