$n$ there exists an interval $[0,t_n]$ in which $v^n_i$ is Hölder continuous with a constant independent of $n$. Now we prove that there exists $t>0$, such that for all $n$, we have $t\leq t_n$ and thus in $[0,t]$ all functions $v^n_i$ are uniformly Hölder continuous. There are exactly 3 instances, when we bound $t_n$ from the above:

1.  In the proof of Lemma \[equitech\].

2.  While ensuring that for all $k\in B^0_i$ we have $|v^n_k-v^n_i|>0$ in $[0,t_n]$.

3.  While ensuring that for all $k\in B^0_i$ the function $\xi$ is positive in $(0,t_n]$.

If each of these bounds from above can be bounded from below by a constant independent of $n$, then so can be $t_n$ for all $n$.

1.  In Lemma \[equitech\], $t_n$ was the supremum of all times $s_n$, such that for all $s\in(0,t_n]$ conditions (\[war1\]) and (\[war3\]) are satisfied. However from step 1 we may estimate $t_n$ better than we could in the proof of Lemma \[equitech\]. We have $$\begin{aligned}
    |r_n(s)|\leq 2L|s|^{1-\alpha}\ \ \ {\rm and}\ \ \ |o_n(s)|\leq 2L|s|^{2-\alpha},\end{aligned}$$ thus by taking $$\begin{aligned}
    \label{tn1}
    t_0:=\left(\frac{1}{10L}|v^n_k(0)-v^n_i(0)|\right)^{\frac{1}{1-\alpha}}\end{aligned}$$ we ensure that (\[war2\]) and consequently (\[war1\]) is satisfied. With the same $t_0$ we obtain also condition (\[war3\]).

2.  For $k\in B^0_i$ 