How to prove that there exists $C_1>0$ such that $ \max\{\|f_{n+1}-f_n\|_{\infty}, \|g_{n+1}-g_n\|_{\infty}\}\le 2^nC^nC_1^n\frac{T^n}{n!} $? I have two iterated update upper bound for two sequences of continuous functions $\{f_n\}$ and $\{g_n\}$, that is there exists $C>0$ such that
$$
\|f_{n+1}-f_n\|_{\infty}\le C\int_0^T|f_n(s)-f_{n-1}(s)|ds +C\int_0^T|g_n(s)-g_{n-1}(s)|ds\\ \le CT[\|f_n-f_{n-1}\|_{\infty}+\|g_n-g_{n-1}\|_{\infty}]
$$
and similarly,
$$
\|g_{n+1}-g_n\|_{\infty}\le C'T[\|f_n-f_{n-1}\|_{\infty}+\|g_n-g_{n-1}\|_{\infty}]
$$
How to prove that there exists $C_1>0$ such that
$$
\max\{\|f_{n+1}-f_n\|_{\infty}, \|g_{n+1}-g_n\|_{\infty}\}\le 2^nC^nC_1^n\frac{T^n}{n!}\,?
$$
Then as $n\to \infty$, $$\max\{\|f_{n+1}-f_n\|_{\infty}, \|g_{n+1}-g_n\|_{\infty}\}\to 0.$$
 A: I will try to use induction to show the conclusion holds for each fixed $T > 0$. To specify different time interval, I will use $\|\cdot\|_\infty^s$ to denote the supremum norm on $[0,s]$. We will see $C_1$ could be set as $\max\{1, \frac{C'}{C}\}$.
For $n=1$, $f_0$ and $f_1$ should satisfy
\begin{equation}
\max\{\|f_1- f_0\|_\infty^s, \|g_1- g_0\|_\infty^s\} \leq 1\text{ for } s > 0
\end{equation}
I think this initial condition should be an assumption.
Now suppose the conclusion holds for $n$:
\begin{equation}
\max\{\|f_n- f_{n-1}\|_\infty^s, \|g_n- g_{n-1}\|_\infty^s\} \leq 2^{n-1}C^{n-1}C_1^{n-1}\frac{s^{n-1}}{(n-1)!}\text{ for } s > 0
\end{equation}
Then for any $t > 0$,
\begin{align}
\|f_{n+1}- f_n\|_\infty^t \leq& C\int_0^t |f_n(s)- f_{n-1}(s)|ds +  C\int_0^t |g_n(s)- g_{n-1}(s)|ds\\
\leq& C\int_0^t \|f_n- f_{n-1}\|_\infty^t ds +  C\int_0^t \|g_n- g_{n-1}\|_\infty^sds\\
\leq & C\int_0^t  2^{n-1}C^{n-1}C_1^{n-1}\frac{s^{n-1}}{(n-1)!} ds +  C\int_0^t  2^{n-1}C^{n-1}C_1^{n-1}\frac{s^{n-1}}{(n-1)!} ds\\
= &  2^nC^nC_1^{n-1}\frac{t^n}{n!} \leq 2^nC^nC_1^n\frac{t^n}{n!}
\end{align}
The last inequality is due to $C_1 \geq 1$. Also
\begin{align}
\|g_{n+1}- g_n\|_\infty^t \leq& C'\int_0^t |f_n(s)- f_{n-1}(s)|ds +  C'\int_0^t |g_n(s)- g_{n-1}(s)|ds\\
\leq& C'\int_0^t \|f_n- f_{n-1}\|_\infty^t ds +  C'\int_0^t \|g_n- g_{n-1}\|_\infty^sds\\
\leq & C'\int_0^t  2^{n-1}C^{n-1}C_1^{n-1}\frac{s^{n-1}}{(n-1)!} ds +  C'\int_0^t  2^{n-1}C^{n-1}C_1^{n-1}\frac{s^{n-1}}{(n-1)!} ds\\
= &  2^nC^{n-1}C_1^{n-1}C'\frac{t^n}{n!} \leq 2^nC^nC_1^n\frac{t^n}{n!}
\end{align}
The last inequality is due to $C_1 \geq \frac{C'}{C}$.
