prove that every finite poset can be embedded into $(\mathbb{N},\;|) $
I'm confused how to construct such a function $f: X \to N$ or prove this function exists. $|N| > |X|$, so definitely the injection could be achieved. I am thinking about mapping the minimal elements in $X$ to the prime numbers and the smallest one to $1$.
Identify the elements of your finite poset $(P,\le)$ with (distinct) prime numbers, and consider the map $x\mapsto\prod_{p\le x}p$. This works if $(P,\le)$ is at most countable and locally finite, i.e., there are only a finite number of elements below any element.