epsilon/delta definition alternative?

The phrase "any epsilon greater than zero" has always seemed somewhat vague.

Question: is this an equivalent definition?

$\forall\mbox{ Natural Numbers }N>0 \,\,\exists\delta>0\mbox{ s.t. }0<|x−x_0|<δ \implies|f(x)−f(x_0)|<1/N$.

• Yes this is equivalent due to the Archimedean property of the natural numbers. I think we just like our $\varepsilon$. – Cameron Williams Feb 8 '14 at 2:23
• In fact, you can just use every second natural number if you want to be more economical... – copper.hat Feb 8 '14 at 2:27
• And to avoid Greek letters altogether, we can say that for any positive integer $N$ there is a positive integer $Q$ such that whenever whenever $|x-x_0|\lt \frac{1}{Q}$, $\dots$. – André Nicolas Feb 8 '14 at 2:33
• What does second natural number mean? – Hawk Feb 8 '14 at 4:38
• He means you can use only even natural numbers: "$\forall$ even natural numbers $N>0$ …" – Lee Mosher May 2 '14 at 13:07

Another possible alternative to the epsilon, delta definition of continuity is Cauchy's original definition. Namely, $f$ is continuous at $x$ if for every infinitesimal $\alpha$ the difference $f(x+\alpha)-f(x)$ is similarly infinitesimal. See Cours d'Analyse.