Archimedean property usage.

Question

I was reading a solution to a problem and I found this statement: "Let $\epsilon > 0$ then there exists a positive integer $n$ such that $1/n < \epsilon.$"

Is there is an educated justification for this sentence? Is this by Archimedean property? Could anyone explain this to me, please?

Answer 1

Indeed, this follows directly from the Archimedean property of $\mathbb{R}.$ See e.g. Abbott Theorem 1.4.2 (he proves this directly), or Rudin Theorem 1.20 (a) by setting $x = \epsilon$ and $y = 1$.

For completeness, Rudin's statement is:

If $x,y \in \mathbb{R}$ and $x > 0$ then there exists a positive integer $n$ such that $nx > y.$

Answer 2

$\Bbb R $ is Archimedean means that

$$(\forall (x,y)\in (0,+\infty)\times \Bbb R) \;\; (\exists n\in \Bbb N) \;\; : \; nx>y$$

now, you apply this with

$$y=1 \text{ and } x=\epsilon$$

Answer 3

Yes, by the Archimedean property, there exists $n\in\mathbb N$ such that $n\times \epsilon>1$, so then $\dfrac1n<\epsilon$.

Answer 4

You can use the archemedian principal two ways.

If $M = \frac 1\epsilon$ and $d = 1 > 0$ then there is an $n$ so that $n = n*d > M =\frac 1\epsilon$ and so $\frac 1n < \epsilon$.

Or if $M = 1$ and $d = \epsilon$ then there is and $n$ so that $n*\epsilon > 1$ and so $\frac 1n < \epsilon$.