On the one hand, it seems clear by assumption that for any $\tau < \tau^{*}_{j_{0}}$, we have $$(1 -\tau^{*}_{j_{0}})^{\frac{2}{\sigma_{s}}} \leqslant (1-\tau)^{\frac{2}{\sigma_{s}}}.$$ On the other hand, hypothesis on $NS(v_{0,n})$ yields $$(1-\tau)^{\frac{2}{\sigma_{s}}} \, \| NS(v_{0,n})(\tau) \|^{2}_{\dot{H}^s} \leqslant\,\, M^{2}_{c} + \varepsilon_{n}.$$ Therefore, from the above remarks, we get $$\| NS(v_{0,n})(\tau) \|^{2}_{\dot{H}^s} \leqslant\,\, \frac{M^{2}_{c} + \varepsilon_{n}}{ (1 -\tau^{*}_{j_{0}})^{\frac{2}{\sigma_{s}}}} \cdotp$$ Combining the above estimate with (\[equation de reference 2\]), we finally get, after multiplication by the factor $(\tau^{*}_{j_{0}} - \tau)^{\frac{2}{\sigma_{s}}} $, $$\label{equation de reference 3}
\begin{split}
\frac{M^{2}_{c} + \varepsilon_{n}}{ (1 - \tau^{*}_{j_{0}})^{\frac{2}{\sigma_{s}}}} \,\, (\tau^{*}_{j_{0}} - \tau)^{\frac{2}{\sigma_{s}}} \geqslant (\tau^{*}_{j_{0}} - \tau)^{\frac{2}{\sigma_{s}}} \, \| NS(\varphi^{j_{0}})(\tau) \|^2_{\dot{H}^s}  - (\tau^{*}_{j_{0}} - \tau)^{\frac{2}{\sigma_{s}}}\,  |\gamma_{n}^{J}(\tau)|.
\end{split}$$ Notice that $(\tau^{*}_{j_{0}} - \tau)^{\frac{2}{\sigma_{s}}}$ is always less than $1$, which allows us to get rid of it in front of the remaining term $|\gamma_{n}^{J}(\tau)|$. In addition, applying (\[remark c\]) and hypothesis on the sequence $\varepsilon_{n}$, one has $$\begin{split}
\frac{M^{2}_{c} + \varepsilon_{0}}{ (1 -\tau^{*}_{j_{0}})^{\frac{2}{\sigma_{s}}}} \,\, (\tau^{*}_{j_{0}} - \tau)^{\frac{2}{\sigma_{s}}} \geqslant c\, - \, |\gamma_{n}^{J}(\tau)|.
\end{split}$$ We first choose $\tau = \tau_{c}$ such that $\tau_{c} < \tau^{*}_{j_{0}}$ and ${\displaystyle}{ \frac{M^{2}_{c} + \varepsilon_{0}}{ (1 -\tau^{*}_{j_{0}})^{\frac{2}{\sigma_{s}}}} \,\, (\tau^{*}_{j_{0}} - \tau_{c})^{\frac{2}{\sigma_{s}}} = \frac{c}{4}}\cdotp$ Then, we take $J$ and $n$ large enough such that ${\displaystyle}{ |\gamma_{n}^{J}(\tau_{c})| \leqslant \frac{c}{2}}\cdotp$ Therefore, we get 