Let $h$ be a continuous metric of $L^{\operatorname{an}}$. If there are a sequence $\{e_{n}\}$ of positive integers and a sequence $\{h_{n}\}$ of metrics such that $h_{n}$ is a semipositive metric of $(L^{\otimes e_{n}})^{\operatorname{an}}$ for each $n$ and

$$\frac{1}{e_{n}}\log\frac{|\raisebox{1.72218pt}{.}|_{h_{n}}(x)}{|\raisebox{1.72218pt}{.}|_{h^{e_{n}}}(x)}$$

converges to $0$ uniformly as $n\to\infty$, then $h$ is semipositive.