For any integer $n\geqslant 1$, let

$$a_{n}=\inf_{\begin{subarray}{c}s\in H^{0}(X,L^{\otimes n})\\ {\left.{s}\right|_{{Y}}}=l^{\otimes n}\end{subarray}}{\Big(}\log\|s\|_{h^{n}}-n\log\|l\|_{Y,h}{\Big)}.$$

Then the sequence $(a_{n})_{n\geq 1}$ is sub-additive, namely one has $a_{m+n}\leq a_{m}+a_{n}$ for any $(m,n)\in\mathbb{N}_{\geq 1}$. In particular, if for sufficiently positive integer $n$, the section $l^{n}$ lies in the image of the restriction map $H^{0}(X,L^{\otimes n})\rightarrow H^{0}(Y,L|_{Y}^{\otimes n})$, then “$\limsup$” in (3) is actually “$\lim$”.