I have a function $F_n:[0,T] \to \mathbb{R}$ which is Lipshitz continuous. The limit $$\lim_{t \to 0^+}\frac{F_n(t)}{t}=0$$ exists for each fixed $n \in \mathbb{N}$. Furthermore, $|F_n(t)| \leq Ct$ where $C$ doesn't depend on $n$.
I also know that $$\lim_{t \to 0^+}F_n(t) =0\quad\text{uniformly in $n$}.$$
Given these facts, is it true that $$\lim_{t \to 0^+}\frac{F_n(t)}{t} =0\quad\text{uniformly in $n$}.$$
In fact $F_n(t) = \tilde F(x_n,t)$ for some convergent sequence $x_n$ if it makes difference and $\tilde F$ is continuous wrt. both arguments.
My attempt is: I know that the product of uniformly convergent sequences is not in general uniformly convergent but I am desparate for a positive answer
Consider $F_n(t) = t^{1+1/n}$ for $t\in [0,1]$. Then, $F_n(t) \le t$ and $F_n(t) / t = t^{1/n} \to 0$ for a fixed $n$ and $t\to 0$. But $$ \sup_{n} \frac{F_n(t)}{t} = 1 \not\to 0.$$