For a positive number $\epsilon>0$, choose a positive integer $n$ such that

$$e^{-\epsilon e_{n}/3}h^{e_{n}}\leq h_{n}\leq e^{\epsilon e_{n}/3}h^{e_{n}}.$$

As $h_{n}$ is semipositive, by Corollary 3.11, there is a positive integer $m$ such that, for all $x\in X^{\mathrm{an}}$, we can find $s\in H^{0}(X,L^{\otimes me_{n}})_{\hat{\kappa}(x)}\setminus\{0\}$ with $\|s\|_{h_{n}^{m},\hat{\kappa}(x)}\leq e^{me_{n}\epsilon/3}|s|_{h_{n}^{m}}(x)$, so that

$$\|s\|_{h^{me_{n}},\hat{\kappa}(x)}\leq e^{\epsilon me_{n}/3}\|s\|_{h_{n}^{m},\hat{\kappa}(x)}\leq e^{2me_{n}\epsilon/3}|s|_{h_{n}^{m}}(x)\leq e^{me_{n}\epsilon}|s|_{h^{me_{n}}}(x).$$

Therefore, the assertion follows from Corollary 3.11. ∎