# Intro Real Analysis: Closed and Open sets Of R. Help with Problem

• MidgetDwarf
In summary, we have two sets, A and B, each with their own limit points. For set A, the limit points are -1 and 1, while for set B, the limit points are the interval [0,1]. This is because for set A, we have two sequences, one for odd values of n and one for even values of n, that converge to -1 and 1 respectively, and since these values are not part of the set, they are considered as limit points. For set B, we have multiple limit points, including 0 and 1, as well as all values in between, due to the Density Theorem of Q in R.
MidgetDwarf
Homework Statement
Let $$A= \{(-1)^n + \frac {2} {n} : n = 1, 2, 3,...\}$$ and $$B =\{x \in ℚ: 0<x<1 \}.$$

What are the limits points of A and B
Relevant Equations
Definition of a limit point : A point x is a limit point of a set A if ## \forall ## epsilon neighborhood of x intersects the set A at some point other than x.

Theorem 1: A point x is a limit point of a set iff ## x= \lim a_n ## for some sequence ##\{a_n)\} ## satisfying ##a_n = x ~\forall n \in N##.

Theorem 2: Density of the Rational Numbers of Q in R.
For the set A:

Note that if n is odd, then ## A = \{ -1 + \frac {2} {n} : \text{n is an odd integer} \} ## . If n is even, A = ## \{1 + ~ \frac {2} {n} : \text{ n is an even integer} \} ## .
By a previous exercise, we know that ## \frac {1} {n} ## -> 0. Let ## A_1 ## be the sequence when n is odd and ## A_2 ## be the sequence when n is even. By the Algebraic Limit Theorem, Lim ## A_1 ## = -1 and Lim ## A_2 ## = 1. Since -1 is not an element of A, then -1 is a limit point of A. Since 1 is never a term of ## A_2 ## , then 1 is a limit point of A. ( By Theorem 2).

Therefore, the limit points of A are -1 and 1.

For the set B:

I know that the set of Limit points of Q is R. Since we are only working with members of Q in the set B. I know that the following two sequences ## \frac {1} {n}## where n is equal to or greater than 2 and ## \frac {n} {n+1} ## reside in B, and they converge to 0 and 1, respectively. Since both 0 and 1 are not members of their respective sequences, then 0 and 1 are limit points of B.

Do I have this correct so far?

But I am really unsure of B. Since by the Density Theorem of Q in R we know that for every real number there exist a sequence of rational numbers that converge to y. So by this Theorem, the limit points of B is the interval [0,1] ?

Sorry for the sloppy LaTex. This is my first time using it.

Last edited by a moderator:
sorry. I tried fixing the code, but i still cannot locate the problem. Wondering if a member can help me also with the syntax.

I think you should have a double '#' in "never a term of # A_2".
The single '#' did not turn on LaTex and things went wrong from there on.

From what I can see, your logic is correct.

It does not matter for a limit point whether it is part of the set or not.
Thus there are more limit points in B than the two you mentioned, and your wording for set A should be corrected accordingly.

MidgetDwarf said:
I tried fixing the code, but i still cannot locate the problem. Wondering if a member can help me also with the syntax.
I fixed things. Take a look at our tuturial -- the link to it is in the bottom left corner of the input pane.

In your statement of theorem 1, did you mean ##a_n \ne x## for all ##n \in \mathbb{N}##?

fresh_42 said:
It does not matter for a limit point whether it is part of the set or not.
Thus there are more limit points in B than the two you mentioned, and your wording for set A should be corrected accordingly.

Yes thank you. I wrote it in a weird way and I think that is what Mark mentioned. I tried to say that, since 1 is not an element of the sequence that converges to 1. Then the conditions you listed are automatically satisfied.

I could have just listed what you wrote and be done. Thank you .

Mark44 said:
I fixed things. Take a look at our tuturial -- the link to it is in the bottom left corner of the input pane.

Thank you. For some reason I missed the tutorial when typing.

fresh_42 said:
It does not matter for a limit point whether it is part of the set or not.
Thus there are more limit points in B than the two you mentioned, and your wording for set A should be corrected accordingly.

Aww. Thank you. I some how mixed some definitions up. Ie., closed. Where A set is called closed if it contains its limit points.

## 1. What is the difference between closed and open sets in Real Analysis?

Closed sets in Real Analysis are sets that contain all of their limit points. This means that if you take any sequence of points within the set, the limit of that sequence will also be within the set. On the other hand, open sets do not contain all of their limit points. This means that there is at least one point in the set that is not a limit point.

## 2. How are closed and open sets related to each other in Real Analysis?

Closed and open sets are complements of each other in Real Analysis. This means that if a set is closed, its complement is open and vice versa. Additionally, any set can be written as a union of a closed set and an open set.

## 3. How do you determine if a set is closed or open in Real Analysis?

To determine if a set is closed or open in Real Analysis, you can use the definition of a limit point. If all limit points of a set are contained within the set, then it is closed. If there is at least one limit point that is not contained within the set, then it is open. Another way to determine this is by using the concept of closure. A set is closed if and only if it is equal to its closure.

## 4. Can a set be both closed and open in Real Analysis?

No, a set cannot be both closed and open in Real Analysis. This is because a set is either closed or open by definition. However, a set can be neither closed nor open, in which case it is called a partially open set.

## 5. How can understanding closed and open sets in Real Analysis help with problem-solving?

Understanding closed and open sets in Real Analysis is important for problem-solving because it allows you to determine if a given set is closed, open, or partially open. This can help in proving theorems and solving problems involving limit points and closure. Additionally, understanding these concepts can lead to a deeper understanding of more advanced topics in Real Analysis.

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