I'm new to topology (baby rudin) and playing around with ways to construct subsets of $R^k$ that have finite numbers of limit points, and I'm wondering if this works:
Let $E_{_0}=[0,1]$.
Remove the half open interval $(\frac{1}{2},1]$, and let $E_{_1}=[0,\frac{1}{2}]$.
Remove the half open interval $(\frac{1}{4},\frac{1}{2}]$, and let $E_{_2}=[0,\frac{1}{4}]$
Continue in this way to construct a sequence of compact sets {$E_{_n}$} such that $E_{_0}\supset E_{_1}\supset E_{_2}\supset ...$ and let $S$ be the intersection of every set in {$E_{_n}$}.
Is $0$ a limit point of $S$?
I see that I could remove segments instead of half open intervals and end up with an infinite set with a limit point only at $0$, I'm just wondering if the above process leads to the logical conclusion that $0$ is an isolated point or if it's still a limit point.
The set $S$ only contains the point $0$. A finite set of reals has no limit point.