(x'\cdot\zeta+{\varepsilon}), ty_n\big)}{t^\sigma y_n^\sigma}
=
\lim_{y_n\searrow0} \frac{t^\beta v(
x'\cdot\zeta+{\varepsilon}, y_n)}{t^\sigma y_n^\sigma}\\&&\qquad=
t^{\beta-\sigma}\psi(x')=t^{\beta-\sigma}\psi_0(x'\cdot\zeta)\end{aligned}$$ Taking $m$ derivatives in $t$ of this identity, we conclude that $$\label{evrt} 
\psi_0^{(m)}\big(t (x'\cdot\zeta+{\varepsilon})-{\varepsilon}\big)\,
(x'\cdot\zeta+{\varepsilon})^m
=\prod_{i=0}^{m-1}(\beta-\sigma-i)\;t^{\beta-\sigma-m}\,\psi_0(x'\cdot\zeta).$$ In addition, by , we have that $D^\gamma\psi(x')=
\zeta^\gamma\psi_0^{(|\gamma|)}(x'\cdot\zeta)$. Hence, evaluating  at $t:=1$ and $x':=0$, if $m=|\gamma|$ we have that $$\label{5379x34} \begin{split}&{\varepsilon}^m D^\gamma\psi(0)=
\zeta^\gamma\psi_0^{(m)}(0)\,{\varepsilon}^m
=\zeta^\gamma \kappa_m\,\psi_0(0)
=\zeta^\gamma \kappa_m\,\psi(0),\\
{\mbox{where }}\qquad&\kappa_m:=\prod_{i=0}^{m-1}(\beta-\sigma-i)
.\end{split}$$ Plugging this information into , and recalling , we find that $$\label{345732}\begin{split}&
0=\frac1{\tilde\Lambda}\sum_{m=1}^k
\sum_{{\gamma\in{\ensuremath{\mathbb{N}}}^{n-1}}\atop{|\gamma|= m}}
\omega_\gamma\,
D^\gamma\psi(0)=\frac1{\tilde\Lambda}
\sum_{m=1}^k
\sum_{{\gamma\in{\ensuremath{\mathbb{N}}}^{n-1}}\atop{|\gamma|= m}}
\kappa_m\,{\varepsilon}^{-m}
\omega_\gamma \zeta^\gamma\psi(0)\\&\qquad\qquad=\frac1{\tilde\Lambda}
\sum_{{\gamma\in{\ensuremath{\mathbb{N}}}^{n-1}}\atop{|\gamma|{\leqslant}k}}
\kappa_{|\gamma|}\,
\omega_\gamma \left(\frac{\zeta}{{\varepsilon}}\right)^\gamma\psi(0)=
\sum_{{\gamma\in{\ensuremath{\mathbb{N}}}^{n-1}}\atop{|\gamma|{\leqslant}k}}
\kappa_{|\gamma|}\,
\omega_\gamma \left(\frac{\zeta}{{\varepsilon}}\right)^\gamma
.\end{split}$$ Since $\frac{\zeta}{\varepsilon}$ ranges in an open set of ${\ensuremath{\mathbb{R}}}^{n-1}$, we deduce from  and the Identity Principle for polynomials that $\kappa_{|\gamma|}\,
\omega_\gamma=0$ for all $\gamma\in{\ensuremath{\mathbb{N}}}^{n-1}$ with $|\gamma|{\leqslant}k$. As a consequence, by , we find that $
\omega_\gamma=0$ for all $\gamma\in{\ensuremath{\mathbb{N}}}^{n-1}$ with $|\gamma|{\leqslant}k$, hence $\omega=0$. This is in contradiction with  and thus we have proved the desired result.

With this, we are in the position of completing the proof of Theorem \[FLFL\] by arguing as follows:

\[89-ppA\] Since the claims in Theorem \[FLFL\] have a linear structure in $f$, $f_{\varepsilon}$ and $u_{\varepsilon}$, by the Stone-Weierstra[ß]{} Theorem, it is enough to prove Theorem \[FLFL\] if $f$ is a monomial. Hence, we fix ${\varepsilon}\in(0,1)$, 