(N^{s-1} \eta_{1} A^{-1})^{\alpha} \right) (1+
\beta)
\end{array}$$

provided that $(N,s)$ satisfies

$$\begin{array}{l}
\frac{ 2 (C_{E} N^{2(1-s)} A^{\frac{p+1}{2}})^{\frac{p-1}{2}} }{\beta N^{\frac{5-p}{2}-}} << 1
\end{array}
\label{Eqn:Cond6}$$

\[prop:EstSmallNrj\]

Let $(C,Q) \in (\mathbb{R}^{+})^{2}$. Let $C_{0} >> 1$ and $N:= \max{( N_{1},N_{2},N_{3},N_{4},N_{5})}$ with

$$\begin{array}{l}
N_{1}  := C_{0} (A^{-1} \eta_{1})^{\frac{2(p-2) \alpha}{5-p}+} A^{\frac{p+2}{5-p}+}
\end{array}$$

$$\begin{array}{l}
N_{2} := C_{0} A^{\frac{p^{2}-1}{5-p}+}
\end{array}$$

$$\begin{array}{l}
N_{3} := C_{0} (X_{C,Q})^{\frac{4(p-1)}{5-p}+} A^{\frac{p(p+1)}{5-p}+}
\end{array}$$

$$\begin{array}{l}
N_{4} := C_{0} ( \bar{C}^{\bar{C} X_{C,Q}} A^{\frac{p(p+1)}{2}} X_{C,Q})^{ \max{ \left( \frac{1}{s_{c} -\frac{1}{2}}+, \frac{1}{1-s_{c}}+
\right)}}
\end{array}$$

and

$$\begin{array}{l}
N_{5} := C_{0} \left( (2 X_{C,Q})^{2(p-1)-} 2 (C_{E} A^{p+1})^{\frac{p+1}{2}} \epsilon_{0}^{-1} (\eta_{1} A^{-1})^{- \alpha}
\right)^{\frac{2}{5-p}+}
\end{array}$$

Let $\tilde{s}:=\tilde{s}(A,C,Q,p) < 1 $ be such that $N^{1-s} \sim 1$ for $s \geq \tilde{s}(A,C,Q,p)$ [^6]. Notice that, for this range of $s$ and for this value of $N$, (\[Eqn:Cond1\]), (\[Eqn:Cond2\]), (\[Eqn:Cond3\]), (\[Eqn:Cond4\]) and (\[Eqn:Cond6\]) are satisfied ( if $C_{0}$ is large enough). Observe also that the following condition is satisfied

$$\begin{array}{ll}
\frac{( 2 X_{C,Q} )^{2(p-1)-} 2 \left( C_{E} N^{2(1-s)} A^{p+1} \right)^{\frac{p-1}{2}}  }   { \beta N^{\frac{5-p}{2}-}} & << 1
\end{array}
\label{Eqn:NControlVar}$$

since $N \geq N_{5}$. Next we define the following statement of induction for $(N,s)$ prescribed above:

$\mathcal{P}_{C,Q}(l)$ is the following: For $i \geq l$, let $\mathcal{C}_{i}$ be the following set

$$\begin{array}{ll}
\mathcal{C}_{i} & = \left\{ w_{i}, \,
