TITLE: why $N>\frac{1}{\epsilon}$?
QUESTION [5 upvotes]: 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?

REPLY [6 votes]: Archimedes Axiom says that you can always find a "big enough" $N$ to beat any real number you are given - the natural numbers are unbounded in the reals.
Basically you imagine you are given an $\epsilon$ to beat - normally $\epsilon$ is thought of as small and positive and you need to get below it.
So if we want $\frac 1 n < \epsilon$ we are dealing with small numbers. To use Archimedes in the form you have it we need to be dealing with large ones, so we convert our target to $n > \frac 1 {\epsilon}$.
$n$ is a natural number, $\frac 1 {\epsilon}$ is a real number, and we have got your problem in a shape where Archimedes can be used.