Given a set $S = \left\{\dfrac{1}{a} + \dfrac{1}{b} : \text{ where } a, b \in \mathbf{N}\right\}\;.$ How could I find the limit points of this set?
My idea is to consider as $a \rightarrow \infty$ and $b \rightarrow \infty$, the sum is equal to $0$. So the first limit point is $0$, but I feel like I'm computing a limit with respect to two variables rather than finding a limit point. I'm a bit confused between limit point and limit.
Limit Point definition
A number $a$ is a limit point of a set $S \subset \mathbf{R} $ if for every $\epsilon > 0$ there is $x \in S$ such that $0 < |a - x| < \epsilon$, that is the set $S \cap (a - \epsilon, a + \epsilon) \setminus \{a\}$ is not empty.
This definition is almost identical to the definition of limit ($\rightarrow \infty$). So in order show $0$ is a limit point, I have to show that for every $\epsilon > 0$, then for there exists a $N = \max(a, b)$, such that $0 < \bigg|0 - \bigg(\dfrac{1}{N} + \dfrac{1}{N}\bigg)\bigg| < \epsilon$
I wonder is this approach reasonable? Any idea would be greatly appreciated.