For every $n>1$, let $A_n = \{\frac{1}{n}+\frac{1}{kn(n-1)}\ : k \in \mathbb{Z}^{+}\}$. What is $$\left(\bigcup_{n=2}^{\infty} A_n \right)'?$$
My initial thought that it should be $\{0\}\cup\{\frac{1}{n}: n \in \mathbb{Z}^{+}\backslash\{1\}\}$, and I can see why irrationals can't be limit points of the set, but what about other rationals? Now I'm not so sure.
Note: $A'$ denotes the set of all limit points of $A$.