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\begin{document}

\maketitle

\pagebreak

\begin{homeworkProblem}
    Le but de cet exercice est de démontrer le théorème de Cantor-Bernstein.
    
    Considérons $E$ et $F$ deux ensembles infinis.
    
    Le théorème de Cantor-Bernstein est le suivant:
    \[
        (\exists \Phi : E \longrightarrow F,
        \exists \Psi : F \longrightarrow E,\quad \Phi \text{ injective et } \Psi \text{ injective} )
        \implies E \simeq F
    \]
    
    \begin{enumerate}
        \item Montrer que toute fonction croissante pour $\subseteq$ admet un point fixe. \footnote{\begin{turn}{180} 
       \textbf{Aide:} Considérer l'ensemble $H = \{x\in P(E) \mid x \subseteq f(x)\}$\end{turn}}
       \item Supposons:
        \[
        \exists \Phi : E \to F,
        \exists \Psi : F \to E,\quad \Phi \text{ injective et } \Psi \text{ injective} 
    \]
       On considere la fonction \begin{align*}
\gamma\colon \mathcal{P}(E) & \longrightarrow \mathcal{P}(E)\\
A&\longmapsto E\setminus\Psi(F\setminus\Phi(A)),
\end{align*}
\item Montrer que $\gamma$ est croissante pour $\subseteq$
\item En déduire une bijection de $E$ dans $F$. Conclure.

    \end{enumerate}
\end{homeworkProblem}

\end{document}