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On note $\mathbb{K} := \mathbb{R}\text{ ou }\mathbb{C}$.

\section{Bolzano-Weierstraß}

\[
	\forall u \in \mathbb{K}^\mathbb{N},\ (\exists K \in \mathbb{R}_+,\ |u| \le K) \implies (\exists \phi \in \mathbf{Inc}(\mathbb N,\ \mathbb N),\ \lim_\infty u\circ \phi \in \mathbb R)
\] 

\section{Bornes atteintes}

\[
	\forall a < b \in \mathbb R,\ \forall f \in \mathcal C([a,\ b],\ \mathbb R),\ (\exists K \in \mathbb R_+,\ |f| \le K) \land \left(\exists i,\ s \in D_f,\ \begin{cases}
		f(i) &= \inf f \\
		f(s) &= \sup f
\end{cases}\right)
\] 

\subsection{Sur $\mathbb C$}
\[
	\forall a < b \in \mathbb R,\ \forall f\in \mathcal C([a,\ b],\ {\color{blue} \mathbb C}),\ \exists K\in \mathbb R_+,\ |f|\le K
\] 

"Atteindre ses bornes" n'a plus de sens dans $\mathbb C$.

\subsection{Image d'un segment}
\[
	\forall [a,\ b] \subset \mathbb C,\ \forall f\in \mathcal C,\ f^{\to }([a,\ b]) \subset \text{segments}(\mathbb C)
\] 

\section{Heine}
\[
	\forall a < b \in \mathbb R,\ \mathcal C([a,\ b],\ \mathbb R) \subset \mathcal{UC}
\] 

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