Converging sequence of solution to a differential equation I was working on a problem about a sequence of functions, each of which is a solution to a sequence of differential equations, that converges to a function which is supposed to be the limit of the previous sequence I mentioned.
The problem is as follows

Let $R=\lbrack t_0-a,t_0+a\rbrack\times\lbrack x_0-b,x_0+b\rbrack$, Suppose $(f_n)_{n\in\Bbb N}$ is a sequence of functions defined in $R$ which converges uniformly to $f$. This function is Lipschitz with respect to $x$ and continuous with respect to $t$. 
Consider the following differential equations
  $$
\begin{cases}
x'(t)=f(t,x)\\
x(t_0)=x_0
\end{cases}
\quad
\begin{cases}
x'(t)=f_n(t,x)\\
x(t_0)=x_n
\end{cases}
$$
Let $\varphi,\varphi_n$ be solutions to both differential equations respectively defined on a common subinterval $\lbrack c,d\rbrack\subseteq \lbrack t_0-a,t_0+a\rbrack$. Show that if $x_n\xrightarrow[]{}x_0$ then $\varphi_n\to\varphi$ uniformly in $\lbrack c,d\rbrack$.

What I've done is to try and follow the definitions of uniform convergence for a sequence of functions and I can't quite grasp how to get to the convergence of $\varphi_n$. I know that the Lipschitz condition on $f$ guarantees the existence of such solutions to the differential equations.
It would seem that the problem has a bit of tricks behind it's back. I'm not quite sure on how should I work with this problem, any help is very much appreciated.
 A: You are right that there are some tricks.  The first (a standard one) is that we write the solutions of the IVPs as solutions of the integral equations
$$
\varphi(t) = x_0 + \int\limits_{t_0}^{t} f(s, \varphi(s)) \, ds, \quad \varphi_n(t) = x_n + \int\limits_{t_0}^{t} f_n(s, \varphi_n(s)) \, ds, \quad \forall{t \in [c, d]}.
$$
Now we need to group terms in a suitable way:
$$
\varphi_n(t) - \varphi(t) = \biggl( \Bigl( x_n - x_0 \Bigr) + \int\limits_{t_0}^{t} (f_n(s, \varphi_n(s)) - f(s, \varphi_n(s)))\, ds \biggr) \\
+ \int\limits_{t_0}^{t} (f(s, \varphi_n(s)) - f(s, \varphi(s))) \, ds,
$$
which gives
$$
\lvert \varphi_n(t) - \varphi(t) \rvert \le \biggl( \lvert  x_n - x_0 \rvert + \biggl\lvert \int\limits_{t_0}^{t} \lvert f_n(s, \varphi_n(s)) - f(s, \varphi_n(s)) \rvert \, ds \biggr\rvert \biggr) \\+ \biggl\lvert\int\limits_{t_0}^{t} \lvert f(s, \varphi_n(s)) - f(s, \varphi(s))\rvert \, ds \biggl\rvert.
$$
Take $\varepsilon > 0$.  For $n$ sufficiently large, $\lvert x_n - x_0 \rvert < \varepsilon$.  Similarly, by the uniform convergence of $f$ to $f_n$ on $R$, we have, for sufficiently large $n$, 
$$
\biggl\lvert \int\limits_{t_0}^{t} \lvert f_n(s, \varphi_n(s)) - f(s, \varphi_n(s)) \rvert \, ds \biggl\rvert < \varepsilon.
$$
Consequently,
$$
\lvert \varphi_n(t) - \varphi(t) \rvert \le 2 \varepsilon + L \biggl\lvert \int\limits_{t_0}^{t} \lvert \varphi_n(s) - \varphi(s) \rvert \, ds \biggl\rvert,
$$
where $L$ is a Lipschitz constant for $f$.  We apply (a simple form of) Grönwall's inequality (another trick) to obtain
$$
\lvert \varphi_n(t) - \varphi(t) \rvert \le 2 \varepsilon e^{L \lvert t - t_0 \rvert} \qquad \forall{t \in [c, d]},
$$
hence
$$
\lvert \varphi_n(t) - \varphi(t) \rvert \le 2 \varepsilon e^{L a} \qquad \forall{t \in [c, d]},
$$
which concludes the proof.
Observe that we do not assume that $f_n$ are Lipschitz.  Indeed, we do not need the uniqueness of the solutions to 
$$
\begin{cases}
x'(t)=f_n(t,x)\\
x(t_0)=x_n
\end{cases}
$$
in our proof.
