Credit for the problem goes to Baby Rudin, Chapter 2, Exercise 5.
We are to construct a bounded set of real numbers with exactly three limit points. Seeing as there are few "computation"-tasks in the book, I haven't really had a chance to build up intuition the way I am used to, however, loeoking at the text in Topology, it seems like I have to get used to this.
I let $X$ be the metric space $\mathbf{R}$. I let $E$ be its subset $\{(1-\epsilon,1+\epsilon),(2-\epsilon,2+\epsilon),(3-\epsilon,3+\epsilon)\}$ for small $\epsilon > 0$. Clearly, $1, 2$ and $3$ are limit points of $E$. However, if I have understood the concept of a limit point (and a neighborhood) correcly, so is $1+\frac{\epsilon}{2}$. I'm having a hard time seeing how I am to limit the number of limit points (no pun intended.)
Best Answer
Take the set $$\left\{\sin{2\pi n\over3}+{1\over n}\ \biggm|\>n\in{\mathbb N}_{\geq1}\right\}\ .$$