%\renewcommand{\theenumi}{\Roman{enumi}}
%\renewcommand{\labelenumiv}{\textbf{\theenumiv.}}

\documentclass{article}%
\usepackage{amssymb}
\usepackage[french]{babel}
\usepackage{geometry}
\usepackage{multicol}
\usepackage{lastpage}
\usepackage[french]{layout}
\usepackage{points}
\usepackage{boxedminipage}
\usepackage{fancybox}
\usepackage{sectsty}
\usepackage{theorem}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{fancyhdr}
\usepackage{afterpage}%
\setcounter{MaxMatrixCols}{30}%
\usepackage{graphicx}
%TCIDATA{OutputFilter=latex2.dll}
%TCIDATA{Version=5.00.0.2552}
%TCIDATA{CSTFile=40 LaTeX article.cst}
%TCIDATA{Created=Tuesday, March 02, 2004 00:11:40}
%TCIDATA{LastRevised=Thursday, September 02, 2004 18:51:30}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="SaveForMode" CONTENT="1">}
%TCIDATA{<META NAME="DocumentShell" CONTENT="Articles\SW\DevoirJFil">}
\geometry{
a4paper,
marginpar=2cm,
lmargin=0cm,
rmargin=2cm,
marginparsep=0pt,
includemp=true,
reversemp=true,
headsep=-0.2in
}
\setlength{\theorempostskipamount}{2\baselineskip}
\theoremheaderfont{\large\bfseries}
{  \theorembodyfont{\large\upshape\mdseries}
\theoremstyle{break}
\newtheorem{theorem}{Th�or�me}
}
\newcounter{Cack}[section]
{   \theorembodyfont{\large\upshape}
\theoremstyle{break}
\newtheorem  {acknowledgement}[Cack]{A savoir}
}
\newcounter{Calg}[section]
{  \theorembodyfont{\upshape}
\theoremstyle{break}
\newtheorem{algorithm}[Calg]{M�thode}
}
\newcounter{Caxi}[section]
{   \theorembodyfont{\itshape}
\newtheorem{axiom}[Caxi]{Axiome}
}
\newcounter{Ccas}[section]
{   \theorembodyfont{\upshape}
\newtheorem{case}[Ccas]{Cas}
}
\newcounter{Ccorr}[section]
\renewcommand{\theCcorr}{\Roman{Ccorr}}
{   \theorembodyfont{\upshape}
\theoremstyle{break}
\newtheorem{claim}[Ccorr]{Correction de l'exercice}
}
\newcounter{Cconc}[theorem]
{   \theorembodyfont{\large\mdseries}
\newtheorem{conclusion}[Cconc]{Conclusion}
}
\newcounter{Ccond}[theorem]
{  \theorembodyfont{\large\mdseries}
\newtheorem{condition}[Ccond]{Hypoth�se}
}
\newcounter{Cconj}[section]
{   \theorembodyfont{\upshape}
\newtheorem{conjecture}[Cconj]{Conjecture}
}
\newcounter{Ccor}[section]
{   \theorembodyfont{\upshape}
\newtheorem{corollary}[Ccor]{Corollaire}
}
\newcounter{Ccrit}[section]
{   \theorembodyfont{\upshape}
\newtheorem{criterion}[Ccrit]{Crit�re}
}
\newcounter{Cdef}[section]
\newcounter{Cnot}[section]
{   \theorembodyfont{\large\upshape}
\theoremstyle{break}
\newtheorem{definition}[Cdef]{D�finition}
\newtheorem{notation}[Cnot]{Notation}
}
\newcounter{Cexa}[section]
{   \theorembodyfont{\slshape}
\newtheorem{example}[Cexa]{Exemple}
}
\newcounter{Cexe}[section]
\renewcommand{\theCexe}{\Roman{Cexe}}
{   \theorembodyfont{\upshape}
\theoremstyle{break}
\newtheorem{exercise}[Cexe]{Exercice}
}
\newcounter{Clem}[section]
{   \theorembodyfont{\upshape}
\newtheorem{lemma}[Clem]{Lemme}
}
\newcounter{Cprob}[section]
\renewcommand{\theCprob}
{\Roman{Cprob}.}
{   \theorembodyfont{\upshape}
\theoremstyle{break}
\newtheorem{problem}[Cprob]{Probl�me}
}
{   \theorembodyfont{\large\mdseries}
\newcounter{Cprop}[section]
\theoremstyle{break}
\newtheorem{proposition}[Cprop]{Propri�t�}
}
\newcounter{Crem}[section]
{   \theorembodyfont{\itshape}
\newtheorem{remark}[Crem]{Remarque}
}
\newcounter{Csol}[section]
{   \theorembodyfont{\upshape}
\theoremstyle{break}
\newtheorem{solution}[Csol]{Solution}
}
\newcounter{Csum}[section]
{   \theorembodyfont{\upshape}
\newtheorem{summary}[Csum]{En r�sum�}
}
\newenvironment{proof}[1]
[D�monstration]
{\noindent\textbf{#1:} }
{\ \fbox{cqfd.}
}
\renewcommand{\theenumi}{\arabic{enumi}}
\renewcommand{\theenumii}{\alph{enumii}}
\renewcommand{\theenumiii}{\roman{enumiii}}
\renewcommand{\labelenumi}{\textbf{\theenumi)}}
\renewcommand{\labelenumii}{\textbf{\theenumii)}}
\renewcommand{\labelenumiii}{\textbf{\theenumiii.}}
\newcommand{\LeTitre}{}
\newcommand{\Titre}[1]{\title{#1} \renewcommand
{\LeTitre}{#1}}
\newcommand{\LaClasse}{}
\newcommand
{\Classe}[1]{\renewcommand{\LaClasse} {#1}}
\newcommand{\LaDate}{}
\newcommand
{\Date}[1]{\author{#1}\renewcommand{\LaDate} {#1}}
\newcommand{\Duree}[1]{\date{#1}}
\newcommand{\Letablissement}{}
\newcommand{\Etablissement}[1]{\renewcommand{\Letablissement} {#1}}
\newcommand{\Ladiscipline}{}
\newcommand{\Discipline}[1]{\renewcommand{\Ladiscipline} {#1}}
\renewcommand{\thesection}
{\Alph{section}}
\renewcommand{\thesubsection}
{\thesection\arabic{subsection}}
\renewcommand{\thesubsubsection}
{\thesubsection.\arabic{subsubsection}}
\newenvironment{qcm}
{   \renewcommand{\labelitemi}{{\setlength{\fboxrule}{.05em}\setlength{\fboxsep}{0pt}\fbox{\rule[-0.3em]{0pt}{1em}\hspace{1em}}}}
\begin{itemize}
}{\end{itemize}}
\begin{document}

%%%%%%%%%%b{gedeonInfo}%%%%%%%%%
%%--Document N� 1315015388473--
%%--Corrig� du document N� 1315016925738--
%%%%%%%%%%e{gedeonInfo}%%%%%%%%%


\begin{claim}
\item
\begin{enumerate}
\begin{enumerate}
$u_{1}=\dfrac{2u_{0}+3}{u_{0}+4}=\fbox{$\dfrac{3}{4}$}\quad;\quad u_{2}%
=\dfrac{2u_{1}+3}{u_{1}+4}=$\fbox{$\dfrac{18}{19}.$}

\item Soit $P_{n}$ la propri\'{e}t\'{e}: "$0<u_{n}<1$". D\'{e}montrons par
r\'{e}currence que $P_{n}$ est vraie $\forall n\in%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
^{\star}:$

\begin{itemize}
\item \textbf{Initialisation:}

On a $0<3/4<1$ donc: \fbox{$P_{1}$ est vraie;}

\item \textbf{H\'{e}r\'{e}dit\'{e}:}

Supposons que $P_{n}$ soit vraie pour un certain rang $n\in{%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
}^{\star}:0<u_{n}<1;$

Posons $f(x)=\dfrac{2x+3}{x+4}$ pour $x\in%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+};$ $f$ est d\'{e}rivable sur $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}$ et:%
\[
f^{\prime}(x)=\dfrac{2(x+4)-(2x+3)}{(x+4)^{2}}=\dfrac{5}{\left(  x+4\right)
^{2}}>0\text{ sur }%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}.
\]


Donc $f$ est croissante sur $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}$ et donc:%
\begin{align*}
0  & <u_{n}<1\\
& \Longrightarrow f(0)<f(u_{n})<f(1)\\
& \Longrightarrow3/4<u_{n+1}<1\\
& \Longrightarrow0<u_{n+1}<1
\end{align*}


D'o\`{u}: \fbox{$P_{n+1}$ est vraie.}

\item \textbf{Conclusion:}

On a d\'{e}montr\'{e} que $P_{1}$ est vraie et que si $P_{n}$ est vraie pour
un certain rang $n>0$ alors $P_{n+1}$ est aussi vraie, donc: \fbox{$\forall
n\in{%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
}^{\star},P_{n}$ est vraie.}
\end{itemize}
\end{enumerate}

\item Pour $n\in{%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
,}$ on pose $v_{n}=\dfrac{u_{n}-1}{u_{n}+3}$ (qui existe car $\forall n\in{%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
,}u_{n}\neq-3$). On a alors, $\forall n\in{%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
:}$

$v_{n}\neq0$ ($u_{n}\neq1$) et:%
\begin{align*}
\dfrac{v_{n+1}}{v_{n}}  & =\dfrac{\dfrac{u_{n+1}-1}{u_{n+1}+3}}{\dfrac
{u_{n}-1}{u_{n}+3}}=\dfrac{u_{n+1}-1}{u_{n+1}+3}\times\dfrac{u_{n}+3}{u_{n}%
-1}\\
& =\dfrac{\dfrac{2u_{n}+3}{u_{n}+4}-1}{\dfrac{2u_{n}+3}{u_{n}+4}+3}%
\times\dfrac{u_{n}+3}{u_{n}-1}=\dfrac{\dfrac{u_{n}-1}{u_{n}+4}}{\dfrac
{5u_{n}+15}{u_{n}+4}}\times\dfrac{u_{n}+3}{u_{n}-1}\\
& =\dfrac{u_{n}-1}{5u_{n}+15}\times\dfrac{u_{n}+3}{u_{n}-1}=\dfrac{1}{5}.
\end{align*}


Donc: \fbox{$\left(  v_{n}\right)  _{n\in{%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
}}$ est g\'{e}om\'{e}trique de raison $q=\tfrac{1}{5}$ et comme $-1<q<1,(v_{n}%
)$ converge vers $0.$}

\item Pour tout $n\in{%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
,}$ on a:%
\begin{align*}
v_{n}  & =\dfrac{u_{n}-1}{u_{n}+3}\Leftrightarrow v_{n}(u_{n}+3)=u_{n}-1\\
& \Leftrightarrow u_{n}(v_{n}-1)=-(3v_{n}+1)\\
& \Leftrightarrow u_{n}=-\dfrac{3v_{n}+1}{v_{n}-1}%
\end{align*}


et comme $\underset{n\rightarrow+\infty}{\lim}v_{n}=0$ on a \fbox{$\underset
{n\rightarrow+\infty}{\lim}u_{n}=-\tfrac{1}{-1}=1.$}
\end{enumerate}
\end{claim}


\end{document}