(1) follows from (1) in Lemma 3.13.

(2) Since $\sigma_{\overline{L}^{\otimes(n+n^{\prime})}}(x)\leq\sigma_{\overline{L}^{\otimes n}}(x)+\sigma_{\overline{L}^{\otimes n^{\prime}}}(x)$ for $n,n^{\prime}\in{\mathbb{N}}(L)$ by (2) in Lemma 3.13, the assertion follows from Fekete’s lemma.

(3) and (4) follow from (2) and (3) in Lemma 3.13 together with (2), respectively.

(5) If $n=0$, then the assertion is obvious, so that we may assume that $n\geq 1$. We fix $n_{0}\in{\mathbb{N}}(L)$. Then $n_{0}\in{\mathbb{N}}(L^{\otimes n})$. Thus, by (2),

∎