Suites de fonctions

(\(f_{n}\)) converge uniformément sur \(I\) vers \(f\), donc : \(\forall \varepsilon > 0 \quad \exists N \quad \forall n, p > N \quad \forall x \in I \quad \left| f_{n}(x) - f_{p}(x)\right| < \varepsilon\).

On fait tendre \(x\) vers \(x_{D}\) : \(\forall \varepsilon > 0 \quad \exists N \quad \forall n, p >N \quad \left| \ell n - \ell p\right| < \varepsilon\).

Donc, (\(\ell n\)) est une suite de Cauchy, qui est donc convergente ; soir \(\ell\) sa limite.

\(\ell = \underset{n \rightarrow +\infty}{\textrm{lim}} \left( \underset{x \rightarrow x_{D}}{\textrm{lim}} \left( f_{n}(x) \right)\right)\)

\(\begin{array}{r c l} \left| f(x) - \ell \right| & = & \left| f(x) - f_{n}(x) + f_{n}(x) - \ell n + \ell n - \ell \right| \\ & \leq & \left| f(x) - f_{n}(x) \right| + \left| f_{n}(x) - \ell n \right| + \left| \ell n - \ell \right| \end{array}\)

\(\begin{array}{r l}\forall \varepsilon > 0 & \exists n_{1} / \forall x \in I \quad \forall n \geq n_{1} \quad \left| f(x) - f_{n}(x) \right| < \frac{\varepsilon}{3}~~(\textrm{convergence uniforme de}~~(f_{n}) \\ & \exists n_{2} / \forall n \geq n_{2} \quad \left| \ell n - \ell \right| < \frac{\varepsilon}{3}~~(\textrm{convergence de}~~\ell n) \end{array}\)

\(\forall n \geq \textrm{max} \{n_{1}, n_{2}\} \quad \forall x \in I \quad \left| f(x) - \ell \right| < \frac{2 \varepsilon}{3} + \left| f_{n}(x) - \ell n\right|\)

Mais \(\ell n = \underset{x \in I}{\underset{x \rightarrow x_{D}}{\textrm{lim}}} f_{n}(x)\) ; ainsi :

\(\exists~\delta > 0 \quad \forall x \in I \quad \left| x - x_{D}\right| < \delta \Longrightarrow \left| f_{n}(x) - \ell n \right| < \frac{\varepsilon}{3}\)

Donc : \(\forall \varepsilon > 0 \quad \exists N \quad \exists \delta > 0~~/ ~~\forall x \left\{ \begin{array}{c} x \in I \\ \left| x - x_{D}\right| < \delta \end{array} \right. \Longrightarrow~\left| f(x) - \ell \right| < \varepsilon\)

Ceci prouve que \(\ell = \underset{x \rightarrow x_{D}}{\textrm{lim}} f(x)\).