1. ## Archimedean property

So there is this statement, claiming that N (Natural numbers) is not bounded above in R, which is really easy to prove, however after proving that they drew a conclusion as I saw in my lecture notes,

Conclusion; For every epsilon (I will write epsilon as e because I don't know how else to write it) greater than zero there exist n in N with 0<1/n<e

How did he drew this conclusion from the fact that N is not bounded above in R? From what I understand what this says , for every epsilon in R (that is greater than 0) you can find a natural number n that is bigger than him. Is this logic correct?

2. ## Re: Archimedean property

Originally Posted by davidciprut
Conclusion; For every epsilon (I will write epsilon as e because I don't know how else to write it) greater than zero there exist n in N with 0<1/n<e How did he drew this conclusion from the fact that N is not bounded above in R?
Using LaTeX you can enter [TEX]\forall\varepsilon>0 [/TEX] gives $\forall\varepsilon>0$

If $\mathbb{N}$ is not bounded above then $\frac{1}{\varepsilon}$ is not an upper bound.
So $\exists n\in\mathbb{N}$ such that $n>\frac{1}{\varepsilon}$. Can you see how it works?