$(u_{0,n})$ belonging to  $\dot{H}^s$, with a finite lifespan ${T_{*}}(u_{0,n})$, such that for any  $t < {T_{*}}(u_{0,n})$ : $$\limsup_{t \to {T_{*}}(u_{0})} ({T_{*}}(u_{0,n}) - t)\, \| NS(u_{0,n}) \|^{\sigma_{s}}_{\dot{H}^s} \leqslant M^{\sigma_{s}}_{c} + \varepsilon_{n}.$$ By definition of ${\displaystyle}{\limsup}$, there exists a nondecreasing sequence of time  $t_{n}$, converging to ${T_{*}}(u_{0})$, such that $$\label{limsup tn}
\forall t \geqslant t_{n},\,\, ({T_{*}}(u_{0,n}) - t)\, \| NS(u_{0,n})(t,x) \|^{\sigma_{s}}_{\dot{H}^s} \leqslant M^{\sigma_{s}}_{c} + \varepsilon_{n}.$$ By rescaling, we consider the sequence $$v_{0,n}(y) = \bigl({T_{*}}(u_{0,n}) - t_{n}\bigr)^\frac{1}{2} \, NS(u_{0,n})\bigl( t_{n},({T_{*}}(u_{0,n}) - t_{n}\bigr)^\frac{1}{2}\,y \bigr).$$ and we have $$\label{rescaling donnee initiale}
\begin{split}
 \|v_{0,n} \|^{\sigma_{s}}_{\dot{H}^s} &= \bigl({T_{*}}(u_{0,n})-t_{n}\bigr)\, \|NS(u_{0,n})(t_{n}) \|^{\sigma_{s}}_{\dot{H}^s}. \\
\end{split}$$ By vertue of (\[limsup tn\]), the sequence $(v_{0,n})_{n \geqslant 1}$ is bounded $\bigl(\hbox{by}\,\, {\displaystyle}{M^{\sigma_{s}}_{c} + \varepsilon_{0}}\bigr)$ in the space $\dot{H}^s$. Moreover, such a sequence generates a Navier-Stokes solution, which keeps on living until the time $\tau^* = 1$ and satisfies $$\label{rescaling solution}
\begin{split}
 NS(v_{0,n})(\tau,y)&= \bigl({T_{*}}(u_{0,n})-t_{n}\bigr)\,^\frac{1}{2} \, NS(u_{0,n})\bigl( t_{n} + \tau\,\bigl({T_{*}}(u_{0,n})-t_{n}\bigr)\, ,\bigl({T_{*}}(u_{0,n})-t_{n}\bigr)\,^\frac{1}{2}\,y \bigr).
\end{split}$$ We introduce ${\displaystyle}{\widetilde{t_{n}} = t_{n} + \tau\,\bigl({T_{*}}(u_{0,n})-t_{n}\bigr)\,}$. Notice that, because of scaling, an easy computation yields $$\label{proposition comparative}
(1-\tau)\, \|NS(v_{0,n})(\tau) \|^{\sigma_{s}}_{\dot{H}^s} = \bigl({T_{*}}(u_{0,n}) - \widetilde{t_{n}}\bigr) \, \|NS(u_{0,n})\bigl( \widetilde{t_{n}} \bigr)\|^{\sigma_{s}}_{\dot{H}^s}.$$ As $\widetilde{t_{n}} \geqslant t_{n}$ for any $n$ (by definition of $\widetilde{t_{n}}$) we combine (\[proposition comparative\]) with (\[limsup tn\]) and we get, for any 