# Existence of a limit point implies existence of inifintely many limit points?

• julypraise
In summary: Ah... I think it's okay to disprove the statement. I showed that A' is not empty but finite anyway. Thanks!
julypraise

## Homework Statement

Prove the following statement is true or not:

the statement:
Let $(X,d)$ be a non-empty metric space and $A$ is a non-empty subset of $X$. Then if $A'$ is not empty, then $A'$ 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 $p_{1},\dots,p_{N}$ are limit points of $A$. Since $A$ has a limit point, it is infinte. Now observe that if $p'\in A,p\notin\{p_{1},\dots,p_{N}\}$ then $p'$ is not a limit point of $A$, i.e., $A\cap B(p';r_{p'})=\{p'\}$ for some $r_{p'}>0$. Thus $A$ has a kind of gap inside it.

But from here, I cannot go further. It almost seems that $A'$ finite is consistent.

julypraise said:
But from here, I cannot go further. It almost seems that $A'$ finite is consistent.

Go with that. I can think of a metric space with one limit point. Can you?

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Dick said:
Go with that. I can think of a metric space with one limit point. Can you?

Sorry, I tried but couldn't think of one. Could you give me at least a hint? Like, what kind of metric space is it with what kind metric??

Well, one obvious way to avoid "an infinite number of limit points" is to start with a space that only contains a finite number of points!

HallsofIvy said:
Well, one obvious way to avoid "an infinite number of limit points" is to start with a space that only contains a finite number of points!

Humm.. But, you know, if a set has only finite points then there is no limit point of this set for whatever metric space is talked about. Isn't it?

julypraise said:
Sorry, I tried but couldn't think of one. Could you give me at least a hint? Like, what kind of metric space is it with what kind metric??

Nothing fancy. Pick a subset of the real numbers. They should be discrete but approach a single limit point. Like 0.

Wow, this one right? $\bigcup_{n=1}^{\infty}\{-\frac{1}{n},\frac{1}{n}\}$.

julypraise said:
Wow, this one right? $\bigcup_{n=1}^{\infty}\{-\frac{1}{n},\frac{1}{n}\}$.

Looks generally ok to me. But I think you want to include 0 in your subset. What do you think?

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Dick said:
Looks generally ok to me. But I think you want to include 0 in your subset. What do you think?

Ah... I think it's okay to disprove the statement. I showed that A' is not empty but finite anyway. Thanks!

## What is a limit point?

A limit point is a point within a set of numbers or a function where it is possible for an infinite number of points within that set to be arbitrarily close to it.

## How does the existence of a limit point imply the existence of infinitely many limit points?

If a set of numbers or a function contains a limit point, it means that there are points within that set that are arbitrarily close to that limit point. Since there are infinitely many numbers between any two numbers, there will also be infinitely many points within the set that are arbitrarily close to the limit point, thus implying the existence of infinitely many limit points.

## Why is the existence of infinitely many limit points important?

The existence of infinitely many limit points is important because it allows us to better understand the behavior and properties of a set of numbers or a function. It also helps us to make more accurate predictions and approximations, as well as to identify important characteristics such as continuity and differentiability.

## Can a set of numbers or a function have a limit point without having infinitely many limit points?

Yes, it is possible for a set of numbers or a function to have a limit point without having infinitely many limit points. This can occur if the set or function has a finite number of points that are arbitrarily close to the limit point, such as a finite sequence of numbers or a polynomial function.

## How is the concept of limit points related to the concept of limits?

The concept of limit points is related to the concept of limits in that both involve the behavior of a set of numbers or a function as it approaches a particular point or value. The difference is that a limit point considers the behavior of the points within a set, while a limit focuses on the behavior of the set as a whole.

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