Satz von Dini

Sei $X\subset \subset {{\mathbb {R} }^{n}},\quad {{\left({{f}_{\nu }}\right)}_{\nu \in \mathbb {N} }}:X{\underset {\text{stetig}}{\mathop {\to } }}\,\mathbb {R}$und es gelte 1. ${{f}_{\nu }}\left(x\right)\leq {{f}_{\nu +1}}\left(x\right)\quad \forall \nu \in \mathbb {N} ,\forall x\in X$ 2. $\exists f\left(x\right):={\underset {\nu \in \mathbb {N} }{\mathop {\sup } }}\,{{f}_{\nu }}\left(x\right)\forall x\in X$