By (1), one has $a_{n}\geq 0$ for any integer $n\geq 1$. Moreover, $a_{n}<+\infty$ if and only if $l^{n}$ lies in the image of the restriction map $H^{0}(X,L^{\otimes n})\rightarrow H^{0}(Y,L|_{Y}^{\otimes n})$. To verify the inequality $a_{m+n}\leq a_{m}+a_{n}$, it suffices to consider the case where both $a_{m}$ and $a_{n}$ are finite. Let $s_{m}$ and $s_{n}$ be respectively sections in $H^{0}(X,L^{\otimes m})$ and $H^{0}(X,L^{\otimes n})$ such that ${\left.{s_{m}}\right|_{{Y}}}=l^{\otimes m}$ and ${\left.{s_{n}}\right|_{{Y}}}=l^{\otimes n}$, then the section $s=s_{m}\otimes s_{n}\in H^{0}(X,L^{\otimes(m+n)})$ verifies the relation ${\left.{s}\right|_{{Y}}}=l^{\otimes(n+m)}$. Moreover, one has

$$\|s\|_{h}=\sup_{x\in X^{\mathrm{an}}}|s|_{h}(x)=\sup_{x\in X^{\mathrm{an}}}|s_{m}|_{h}(x)\cdot|s_{n}|_{h}(x)\leqslant\|s_{m}\|_{h}\cdot\|s_{n}\|_{h}.$$

Since $s_{m}$ and $s_{n}$ are arbitrary, one has $a_{m+n}\leq a_{m}+a_{n}$. Finally, by Fekete’s lemma, if $a_{n}<+\infty$ for sufficiently positive integer $n$, then the sequence $(a_{n}/n)_{n\geq 1}$ actually converges in $\mathbb{R}_{+}$. The proposition is thus proved. ∎