In the set $(\mathbb N^{+})^{\mathbb N}$ we have partially ordered set: $$f \le g \Leftrightarrow (\forall n \in \mathbb N) f(n)|g(n).$$ (a) Whether the partially ordered set $\left\langle (\mathbb N^{+})^{\mathbb N}, \le\right\rangle $ has the smallest element?
(b) Let $A=\left\{ f \in (\mathbb N^{+})^{\mathbb N}: (\forall n \in \mathbb N) f(n) \ge 2\right\} $. Prove that set of all minimal elements of $A$ is infinite.
(c) Let $f_{1}: \mathbb N \rightarrow \mathbb N^{+}$ where $f_{1}(n)=n+1$ and $f_{2}: \mathbb N \rightarrow \mathbb N^{+}$ is a function which is constantly equal to $2$ for every $n \in \mathbb N$. Find supremum $\left\{ f_{1}, f_{2} \right\} $ in the partialy ordered set $\left\langle (\mathbb N^{+})^{\mathbb N}, \le\right\rangle $.
My solution:
(a) I think the answer is $0$ because if $f(n)=g(n)$ then remainder of the division $f(n)$ by $g(n)$ is $0$. However I don't know how to prove it.
Moreover I don't have idea how to do other tasks: (b) and (c).
Can you help me and talk me what to look for to learn to do this type of task?