general topology - Construct a set of real numbers whose limit points comprise the set of integers $mathbb{Z}$

My thought process is the following: Let $S=\{ m + \frac{1}{n}| m \in \mathbb{Z},n \in N \}$. Then I need to show that the limit points of $S$ are indeed the integers and that these are the only limit points. I don't know where to go from here.

Answer

Your example is correct because the $\lim_{n\to \infty} m+\frac{1}{n} = m\in \mathbb{Z}$ for all $m\in \mathbb{Z}$, however your trick here is that you use that $m\in\mathbb{Z}$.