i am asking here the most simple and dumbest question ever. once i was reading about convergence yesterday night, i came to the notion which beats me over times: The archimedean axiom
the text was about the convergence of the sequence $\frac{1}{n}$.
The Sequence $\frac{1}{n}$ is a zero-sequence: Proof: There is $\epsilon>0$. According to AA, there is $N\in\Bbb{N}$ with $N>\frac{1}{\epsilon}$. Then $|\frac{1}{n}-0|=\frac{1}{n}<\epsilon$ for all $n\ge N$
what i dont understand is: why AA and why $N>\frac{1}{\epsilon}$? what does that mean in words? i imagined in a line saying that $\epsilon=2$ then $\frac{1}{\epsilon}$ is the 0.5 part of that line. So the $N$ should be greater than that part of $\epsilon$-neighborhood, why?