Using successively the equivalence between the norm $\| \cdot \|_{\frac{1}{2}, \rho}$ and $\| \cdot \|_{\frac{1}{2}, \rho}$, the standard trace theorem from $H^{\frac{1}{2}}(\partial \omega^{\rho})$ to $H^1(\Omega)$ and Theorem 2.2, we finally have

$$ \sum_{k=0}^{\infty} (1+k)(|a_{r,k}^{\varepsilon}|^2 + |a_{\theta,k}^{\varepsilon}|^2 + |b_{r,k}^{\varepsilon}|^2 + |b_{\theta,k}^{\varepsilon}|^2) \lesssim (1+N)^2 \sum_{k=0}^{\infty} (1+k)\xi_k^2 \quad (A.85) \\ \lesssim (1+N)^2 \|\mathbf{u}_N^{\varepsilon}\|_{\frac{1}{2},\rho} \lesssim (1+N)^2 \|\mathbf{u}_N^{\varepsilon}\|_{1,\Omega} \lesssim (1+N)^2 (|\mathbf{v}_b| + \varepsilon|w_b|). $$

This concludes the proof. □

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