First we assume that $h$ is semipositive. By using Proposition 3.10, we can find a positive integer $n$ such that $L^{\otimes n}$ is generated by global sections and

$$|\raisebox{1.72218pt}{.}|_{h^{n}}(x)\leq|\raisebox{1.72218pt}{.}|^{\mathrm{quot}}_{h^{n}}(x)\leq e^{n\epsilon/2}|\raisebox{1.72218pt}{.}|_{h^{n}}(x)$$

for all $x\in X^{\mathrm{an}}$. On the other hand, there is $s\in H^{0}(X,L^{\otimes n})_{\hat{\kappa}(x)}\setminus\{0\}$ such that $\|s\|_{h^{n},\hat{\kappa}(x)}\leq e^{n\epsilon/2}|s|^{\mathrm{quot}}_{h^{n}}(x)$. Thus,

$$\|s\|_{h^{n},\hat{\kappa}(x)}\leq e^{n\epsilon/2}|s|^{\mathrm{quot}}_{h^{n}}(x)\leq e^{n\epsilon}|{s}|_{h^{n}}(x).$$

Next we consider the converse. For a positive integer $m$, there is a positive integer $e_{m}$ such that, for any $x\in X^{\mathrm{an}}$, we can find $s\in H^{0}(X,L^{\otimes e_{m}})_{\hat{\kappa}(x)}\setminus\{0\}$ with $\|s\|_{h^{e_{m}},\hat{\kappa}(x)}\leq e^{e_{m}/m}|s|_{h^{e_{m}}}(x)$. Clearly $L^{\otimes e_{m}}$ is generated by global sections. Moreover,

$$|s|_{h^{e_{m}}}(x)\leq|s|^{\mathrm{quot}}_{(H^{0}(X,L^{\otimes e_{m}}),\|\raisebox{1.20552pt}{.}\|_{h^{e_{m}}})}(x)\leq e^{e_{m}/m}|s|_{h^{e_{m}}}(x),$$

that is,

$$0\leq\frac{1}{e_{m}}\log\left(\frac{|\raisebox{1.72218pt}{.}|^{\mathrm{quot}}_{(H^{0}(X,L^{\otimes e_{m}}),\|\raisebox{1.20552pt}{.}\|_{h^{e_{m}}})}(x)}{|\raisebox{1.72218pt}{.}|_{h^{e_{m}}}(x)}\right)\leq\frac{1}{m}.$$

Thus $h$ is semipositive. ∎