Satz von Dini

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