Cauchy condition states that the limit for a function exists if: $\forall \epsilon > 0 \ \ \exists \delta > 0$ s.t. $\forall x \in E$ and $x \neq a$ and $|x-a| < \delta$ then $|f(x)-A|<\epsilon$.
So take the definition of the function like the following: $f(x) = 1$ for $x \in \mathbb{N}$ and $f(x) = 0$ for $x \notin \mathbb{N}$. I am struggling to apply the Cauchy condition here, simply because the limit of the natural numbers is positive infinity, and thus $a = +\infty$, and there is no such $\delta$ s.t. $|x-\infty| < \delta$. What am I missing here?
EDIT: the limit for when the input to the function is natural numbers.