TITLE: Question about limit points of a Subset of $\mathbb{R}$
QUESTION [7 upvotes]: The question :

Let $D$ be a nonempty subset of the reals that is bounded above. Is the supremum of $D$ a limit point of $D$?

My Reasoning: I think this is false for these two cases.
Case 1:If I look at $D = \{n \in \mathbb{Z} | n \le 0\}$ the supremum is $0$. And since I need a convergent sequence $\{x_n\} \subset D/\{0\}$ the converges to $0$ for it to be a limit point I can say in this case if I look at $\epsilon = \frac{1}{2}$ for the converges of the sequence it will fail to converge and so $0$ isn't a limit point. 
Case 2: Also if I look at $D = {0}$ then the supremum is $0$. And $D$ is a subset of the reals. So if I look for a sequence $\{x_n\} \subset D/\{0\}$ I can't make one because $D/\{0\}$ is the empty set. 
My question is this. Since the problem asked about an arbitrary subset of the reals $D$, can I define $D$ to give a counterexample like above or have I misunderstood the question?
--Thanks in advance.

REPLY [2 votes]: If $D$ is finite of course it does not have any limit points. 
If  the supremum called $s$ does not belong to $D$, then it will be its limit point, if not one can find a neighborhood of $s$ with no point of $D$ which contradicts the supremum assumption. 
Also suppose that $D$ does not have any isolated points. Now if $D$ has a supremum called $s$, pick a neighborhood of $s$. There should be at least a point of $D$, called $d$ which is in this neighborhood otherwise one can find smaller upper bound ($d$ may be equal to $s$). Now because $D$ does not have any isolated points, every neighborhood of $d$ has infinitely many points of $D$. For $d=s$, the supremum has the same property and if not, one can find a neighborhood of $d$ which falls into the neighborhood of $s$ and $s$ is a limit point.