Uniform Convergence and ODE

The following problem appeared on a past exam at my institution:

Suppose that for each $n\in\mathbb{N}$, $u_n:\mathbb{R}\rightarrow\mathbb{R}$ is a differentiable function satisfying $$u_n'(x)=F(u_n(x),x)$$ for all $x\in\mathbb{R}$ where $F:\mathbb{R}^2\rightarrow\mathbb{R}$ is a continuous bounded function.

(i) If $u:\mathbb{R}\rightarrow\mathbb{R}$ is a function such that $u_n\rightarrow u$ uniformly as $n\rightarrow\infty$, prove that $u$ is differentiable with $$u'(x)=F(u(x),x)$$ for all $x\in\mathbb{R}$.

(ii) If $x_0,y_0\in\mathbb{R}$ and $u:\mathbb{R}\rightarrow\mathbb{R}$ is the unique function satisfying $$u'(x)=F(u(x),x), u(x_0)=y_0$$ for all $x\in\mathbb{R}$ and if $u_n(x_0)\rightarrow y_0$ as $n\rightarrow\infty$, prove that $u_n\rightarrow u$ uniformly as $n\rightarrow\infty$.

To prove (i), I've already tried showing that $u_n'\rightarrow g$ uniformly as $n\rightarrow\infty$ where $g:\mathbb{R}\rightarrow\mathbb{R}$ is given by $g(x):=F(u(x),x)$ but to no avail. To prove (ii), I recognize that it suffices to show that $(u_n)_{n\in\mathbb{N}}$ converges uniformly to some function $\tilde{u}:\mathbb{R}\rightarrow\mathbb{R}$ because (i) then implies $\tilde{u}=u$. Does anyone have any suggestions?