If we have a sequence of smooth functions $\{f_{n}\}_{n}$ where $f_{n}: U \rightarrow \mathbb{R}$, where $U \subset \mathbb{R}^{n}$.
We are given the following two results:
For $x \in U$ we have $|f_{n}(x)| < \infty$ for all $n=1,2,...$ and also similarly $|Df_{n}(x)| < \infty$ for all $n=1,2,...$ then how does it follow that $\{f_{n}\}_{n}$ is uniformly bounded and equicontinuous. Note that $Df_{n}$ is the gradient vector.
The uniform boundedness seems to follow directly from $|f_{n}(x)| < \infty$ for all $n=1,2,...$ and all $x$, but I can't see how equicontinuity follows, maybe I'm missing some result that is used?
Thanks for any assistance, let me know if something is unclear.