with $F:=\varepsilon\Delta U_k^\varepsilon$, $F':=\varepsilon'\Delta U_k^{\varepsilon'}$. It follows from that $$\begin{split}
\|U_k^\varepsilon(t')-U_k^{\varepsilon'}(t')\|^2_{H^\sigma}\lesssim\|U_k^\varepsilon(t)-U_k^{\varepsilon'}(t)\|^2_{H^\sigma}+\int_{t}^{t'}&\big(\|U_k^\varepsilon(s)-U_k^{\varepsilon'}(s)\|^2_{H^\sigma}+(\varepsilon+\varepsilon')\|U_k^\varepsilon(s)-U_k^{\varepsilon'}(s)\|_{H^\sigma}\big)\\
&\times\big(1+\|U_k^\varepsilon(s)\|_{H^{\sigma+2}}+\|U_k^{\varepsilon'}(s)\|_{H^{\sigma+2}}\big)\,ds
\end{split}$$ for any $0\leq t\leq t'\leq 1$ and any $\sigma\in[3,K]\cap\mathbb{Z}$. Therefore, using , $$\sup_{t\in [0,1]}\|U_k^\varepsilon(t)-U_k^{\varepsilon'}(t)\|^2_{H^{M+2}}\leq C(k)(\varepsilon+\varepsilon'),$$ which shows that $\lim_{\varepsilon\to 0}U_k^\varepsilon$ exists for any $k$ fixed. To summarize, for any $k\in\mathbb{Z}_+$ we have constructed a solution $U_k\in C([0,1]:H^{M+2})$ of the equation $$(\partial_t+i\Lambda)U_k=\frac{i}{4}\sum_{j=1}^2\Lambda R_j\big[|\nabla|\Lambda^{-1}(U_k+\overline{U_k})\cdot R_j(U_k-\overline{U_k})\big]+\frac{i}{8}\sum_{j=1}^2|\nabla|\big[R_j(U_k-\overline{U_k})\cdot R_j(U_k-\overline{U_k})\big],$$ with $U_k(0)=P_{(-\infty,k]}U_0$. Moreover, see , $$\label{fgh4}
\sup_{t\in[0,1]}\|U_k(t)\|_{H^3}\lesssim \|U_0\|_{H^3},\qquad \sup_{t\in[0,1]}\|U_k(t)\|_{H^{M+d}}\lesssim 2^{dk}\|U_0\|_{H^M},\qquad d\in\{0,1,2\}.$$

We show now that the sequence $\{U_k\}_{k\in\mathbb{Z}_+}$ is Cauchy in $C([0,1]:H^M)$. For this we apply Lemma \[enerdif\] to the solutions $U:=U_k$ and $U':=U_{k'}$, $k+1\leq k'$, with $F=F'=0$. It follows from that $$\|U_{k'}(t')-U_k(t')\|^2_{L^2}\lesssim\|U_{k'}(t)-U_k(t)\|_{L^2}^2+\int_{t}^{t'}\|U_{k'}(s)-U_k(s)\|^2_{L^2}\,ds,$$ for any $0\leq t\leq t'\leq 1$. Since $\|U_{k'}(0)-U_k(0)\|_{L^2}\leq 2^{-3k}$ it follows that $\|U_{k'}(t)-U_k(t)\|_{L^2}\lesssim 2^{-3k}$ for any $t\in[0,1]$. By interpolation with , $$\label{fgh5}
\sup_{t\in[0,1]}\|U_{k'}(t)-U_k(t)\|_{H^{3/2}}\lesssim 2^{-3k/2}.$$ It follows now from and the bound that $$\begin{split}
\|U_{k'}(t')-U_k(t')\|^2_{H^M}&\lesssim\|U_{k'}(t)-U_k(t)\|_{H^M}^2\\
&+(1+\|U_0\|_{H^M})\int_{t}^{t'}\|U_{k'}(s)-U_k(s)\|^2_{H^M}+2^{-k/2}\|U_{k'}(s)-U_k(s)\|_{H^M}\,ds,
\end{split}$$ for any $0\leq t\leq t'\leq 1$. Therefore $$\sup_{t\in[0,1]}\|U_{k'}(t)-U_k(t)\|_{H^M}\leq C(\|U_0\|_{H^M})[2^{-k/2}+\|U_{k'}(0)-U_k(0)\|_{H^M}.$$ This shows that the sequence $\{U_k\}_{k\in\mathbb{Z}_+}$ is Cauchy in $C([0,1]:H^M)$.

Letting $U:=\lim_{k\to\infty}U_k$, we have constructed a solution $U\in C([0,1]:H^M)$ of such that $U(0)=U_0$ and $$\sup_{t\in[0,1]}\|U(t)\|_{H^3}\lesssim \| U_0\|_{H^3}.$$ The inequality follows from the corresponding inequality satisfied uniformly by the functions $U_k^\varepsilon$. The uniqueness of the solution $U$ follows from .

[**[Step 1.]{}**]{} For any integer $J\geq 