In some paper I read, it says we may use the abstract version of the Arzela Ascoli theorem to conclude for some sequence $f_k: I \times \mathbb{R}^n \rightarrow \mathbb{R}^n$, $I$ some interval, that
$f_k(t) \rightarrow f(t) \ \ \ \text{in} \ C_\text{loc}(\mathbb{R}^n),\ \text{uniformly in } t \in I. $
Unfortunately, I don't know such a generalized version of the theorem. Thus I wonder, what has to be shown to conclude this. Is it just uniform boundedness and equicontinuity, both uniformly in $t$? Or does this require some more special conditions? Thank you for any hints.