[Math] Given a number ε > 0, prove there exists a natural number $N$ such that 1/N < ε

Question

I believe there are three cases. I think I have figured out the first one.

Case 1: Let $ε$ = 1 and N > 1. Take $ N$=2. Then 1/2 < $ε$ ≡ 1/2 < 1.

Case 2: Let 0<$ε$<1. I believe that I would use the Archimedian Property, which states that, "If x is a real number, then either x is an integer or there exists an integer n, such that $n$ < $x$ < $n$ + 1.
I am having trouble applying this to case 2.

And then there's the case when ε > 1, but I am confused on how to prove this as well.

Any hints would be much appreciated.

Answer 1

HINT: Given any real number, say $x$, you can a positive integer $N$ so that $N>x$. Now what should you let $x$ be so that when you `flip both sides upside-down', you get the desired inequality?