This is the proof I have written:
Suppose $\bigcap_{n\in \mathbb{N}} (0, \frac{1}n)\neq \emptyset $. Let $S=\bigcap_{n\in \mathbb{N}} (0, \frac{1}n)$. Then let $x\in S$, so $x>0$. By Archimedean Principle we have $\frac{1}N <x$ so $x\notin (o, \frac{1}N)$. So $x\notin \bigcap_{n\in \mathbb{N}} (0, \frac{1}n)$ which is a contradiction. Thus, $\bigcap_{n\in \mathbb{N}} (0, \frac{1}n)= \emptyset $.
My professor said I am very close but I need to prove the so part. It thought that was included with the Archimedean Principle? What else would I have to add to make it a complete proof?