Does every submonoid of $\mathbb N_{\ge 0}$ contained in some numerical semigroup?

Question

Let $A$ be a submonoid of the monoid of non-negative integers $\mathbb N$ under addition. Then does there necessarily exist a submonoid $S$ of $\mathbb N$ such that $A\subseteq S$ and $\mathbb N \setminus S$ is finite but non-empty ?

Answer 1

Yes. Let $a$ be any element of $\Bbb N \setminus A$. Then $$S=A \cup \{ x \in \Bbb N : x>a\}$$ is a monoid containing $A$, and $\Bbb N \setminus S$ is finite and non-empty since $$\{ a\} \subseteq \Bbb N \setminus S \subseteq \{ 0, \dots , a\}$$