\, - \, \eta.$$ Now, choosing $\eta$ small enough (namely ${\displaystyle}{\eta = \frac{\alpha_{0}}{2}}$) we get a contradiction which proves that $NS(\varphi^{j_{0}})$ is a sup-critical solution. This concludes the proof of step $2$ and thus the point (\[prop point 1\]) is proved.\
$\bullet$ Step $3$ : let us prove the point (\[prop point 2\]) of Proposition \[proposition critical element \]. The proof is a straightforward adaptation of the previous one. We shall use that $NS(\varphi^{j_{0}})$ is a sup-critical solution: $$\limsup_{\tau \to 1} (1 - \tau)\,  \| NS(\varphi^{j_{0}})(\tau) \|^{\sigma_{s}}_{\dot{H}^s} =\, M^{\sigma_{s}}_{c}.$$ As we always have ${\displaystyle}{\sup_{\tau < 1}\, (1 - \tau)\,  \| NS(\varphi^{j_{0}})(\tau) \|^{\sigma_{s}}_{\dot{H}^s}   \geqslant \limsup_{\tau \to 1} (1 - \tau)\,  \| NS(\varphi^{j_{0}})(\tau) \|^{\sigma_{s}}_{\dot{H}^s}}$, we get a first inequality : ${\displaystyle}{\sup_{\tau < 1}\, (1 - \tau)\,  \| NS(\varphi^{j_{0}})(\tau) \|^{\sigma_{s}}_{\dot{H}^s}   \geqslant M^{\sigma_{s}}_{c} }$.\
According to the previous computations, we have, for any $\tau < 1$, $$M^{2}_{c} + \varepsilon_{n}  \geqslant (1 - \tau)^{\frac{2}{\sigma_{s}}} \, \| NS(\varphi^{j_{0}})(\tau) \|^2_{\dot{H}^s}\,  -\,   |\gamma_{n}^{J}(\tau)|.$$ Hypothesis on the remaining term $|\gamma_{n}^{J}|$ implies that ${\displaystyle}{\sup_{\tau < 1}\, (1 - \tau)\,  \| NS(\varphi^{j_{0}})(\tau) \|^{\sigma_{s}}_{\dot{H}^s}   \leqslant M^{\sigma_{s}}_{c} },$ which provides the second desired inequality. This ends up the proof of (\[prop point 2\]).\
Let us recall some notation and add a few words 