- #1

julypraise

- 110

- 0

## Homework Statement

Prove the following statement is true or not:

the statement:

Let [itex](X,d)[/itex] be a non-empty metric space and [itex]A[/itex] is a non-empty subset of [itex]X[/itex]. Then if [itex]A'[/itex] is not empty, then [itex]A'[/itex] is infinte.

## Homework Equations

Definition of limit point and its negation.

## The Attempt at a Solution

I tried to prove by contradiction in this way: (to prove the statement is true)

Suppose [itex]p_{1},\dots,p_{N}[/itex] are limit points of [itex]A[/itex]. Since [itex]A[/itex] has a limit point, it is infinte. Now observe that if [itex]p'\in A,p\notin\{p_{1},\dots,p_{N}\}[/itex] then [itex]p'[/itex] is not a limit point of [itex]A[/itex], i.e., [itex]A\cap B(p';r_{p'})=\{p'\}[/itex] for some [itex]r_{p'}>0[/itex]. Thus [itex]A[/itex] has a kind of gap inside it.

But from here, I cannot go further. It almost seems that [itex]A'[/itex] finite is consistent.