Let $d\in\mathbb N$, $(\varphi_n)_{n\in\mathbb N}\subseteq C(\mathbb R^d)$ be uniformly equicontinuous with $\varphi_n(0)=1$ and $\varphi_n>0$ for all $n\in\mathbb N$ and $(f_n)_{n\in\mathbb N}\subseteq C(\mathbb R^d)$ with $$0\le f_n\le-3\ln\varphi_n\tag1\;\;\;\text{for all }n\in\mathbb N.$$
How can we show that
- $\sup_{n\in\mathbb N}\sup_{\overline B_1(0)}f_n<\infty$; and
- $(f_n)_{n\in\mathbb N}$ is equicontinuous at $0$?
Regarding 1.: Since $(\varphi_n)_{n\in\mathbb N}$ is uniformly equicontinuous, there is a $\delta>0$ with $$\forall n\in\mathbb N:\forall x\in B_\delta(0):\varphi_n(x)>\frac12\tag2.$$ Now we somehow need to use $(1)$ ...
Regarding 2.: Let $\varepsilon>0$. Since $(\varphi_n)_{n\in\mathbb N}$ is equicontinuous at $0$ and $\varphi_n(0)=1$ for all $n\in\mathbb N$, there is a $\delta>0$ with $$\forall n\in\mathbb N:\forall x\in B_\delta(0):|\varphi_n(x)-1|<\varepsilon\tag3.$$ How can we use this (and probably $(1)$) to show the claim?