We keep the notation of the previous proposition. By definition the second condition is equivalent to

(4)

$$\liminf_{n\rightarrow+\infty}\frac{a_{n}}{n}=0.$$

Since $L$ is ample, Proposition 1.1 leads to the convergence of the sequence $(a_{n}/n)_{n\geq 1}$ in $\mathbb{R}_{+}$. Hence the condition (4) is equivalent to $\lambda_{h}(l)=0$. ∎