The estimates for the monomials in the lemma can be used to give another proof of the Hausdorff convergence in Proposition 3.2. Note that for any $\epsilon>0$ for large enough $s$, the monomial $x^{v}s^{\lambda(v)}$ become dominant in $\mathrm{Log}_{s}^{-1}(Q^{\lambda}_{v}(\epsilon))$. Hence the equation for $H_{s}^{\operatorname{af{}f}}$ cannot have solutions in this domain. This means that the amoebas $\mathcal{A}^{\lambda}_{s}$ are $\epsilon$-close to their spine $\mathcal{A}^{\lambda}_{\infty}$.