Let $X:=\{g:[0,1]\rightarrow \mathbb{R}\:|\:g\in C^1,|g|\leq 10, |g^{'}| \leq 10\}$. I am interested in showing that $X$ is a relatively compact subset of $C([0,1])=\{g:[0,1]\rightarrow \mathbb{R}\:|\:g \text{ is continuous}\}$ using the norm $||g||_{\text{sup}}:=\sup_{x\in[0,1]}|g(x)|.$
By Arzela-Ascoli theorem, it suffices to show that $X$ is bounded and equicontinuous. It is easy to show that $X$ bounded. But I'm not sure how to proceed with the equicontinuous part. So any help will be very useful.
Thanks.
A set of functions with a common Lipschitz constant is (uniformly) equicontinuous and hence equicontinuous. Here we have for all $g$ that:
$|g(x)-g(y)|=|g'(c)(x-y)|\leq\,10|x-y|$ so there is a common Lipschitz
constant for all $g$.
Therefore the family is uniformly equicontinuous since we can take, given
an $\epsilon$ as $\delta=\dfrac{\epsilon}{10}$.