Does a finite set of single real numbers have any limit points?

[xsw001]

D={set of real numbers consisting of single numbers}
Show set D has no limit points, and show the set of Natural numbers has no limits points.

I know it's a very simple question. I don’t know my way of approaching this is appropriate or not. Let me know. Thanks.

A finite set of real numbers consisting of single numbers is not a sequence and doesn’t converge to a specific number. Therefore can’t have limit points.

I know the fact that the set of Natural numbers are denumerable (infinite countable), and it diverges, therefore natural numbers have no limit point.

 

[HallsofIvy]

What do you mean by "a real number consisting of a single number"?

Perhaps you intended "a set of real numbers which contains a single number". That is not at all "{set of real numbers consisting of single numbers}".

Let \{x_0\} be such a set. What is the definition of limit point of a set? What would a limit point of this set be like?

 

[xsw001]

Yes, you're right.

Well, limit point of a set D\{xo} is a number Xo such that every deleted delta neighborhood of xo contains members of the set. For any delta>0, we can always find a member of the set which is not equal to xo, such that |x-xo|<delta.