MHB Accumulation Points of $(-1)^n + (1/m)$: Is it True?

[Dustinsfl]

All numbers of the form $(-1)^n + (1/m)$, $n,m\in\mathbb{Z}^+$.

Is this true $(-1)^n + (1/m) = (-1,1)$? If so, the accumulation points are $x\in [-1,1]$ and the set is open.

 

[CaptainBlack]

All numbers of the form $(-1)^n + (1/m)$, $n,m\in\mathbb{Z}^+$.

Is this true $(-1)^n + (1/m) = (-1,1)$? If so, the accumulation points are $x\in [-1,1]$ and the set is open.

$3/4 \in (-1,1)$, but for what $n, m \in \mathbb{Z}^+$ is $3/4=(-1)^n+(1/m)$ ?

Moreover for $n$ even for what $m$ is $(-1)^n+(1/m) \in (-1,1)$ ?

CB

 

[Dustinsfl]

If I can't write this an interval, how can I determine if it is open or closed and the accumulation points?

 

[Ackbach]

If I can't write this an interval, how can I determine if it is open or closed and the accumulation points?

There can only be two accumulation points. The $(-1)^{n}\in\{-1,1\}$, and the $(1/m)$ are small positive numbers getting ever smaller.

As for open or closed, you can consider either the set itself or its complement. I would advise looking at the complement, and thinking about a union you could use to write the entire complement. Recall that the union of an arbitrary number (including infinite) of open sets is open. Can you write the complement of your set as an open set? If so, then the set itself must be ...

 

[Dustinsfl]

There can only be two accumulation points. The $(-1)^{n}\in\{-1,1\}$, and the $(1/m)$ are small positive numbers getting ever smaller.

As for open or closed, you can consider either the set itself or its complement. I would advise looking at the complement, and thinking about a union you could use to write the entire complement. Recall that the union of an arbitrary number (including infinite) of open sets is open. Can you write the complement of your set as an open set? If so, then the set itself must be ...

It is open since it doesn't contain its limit points. Correct?

 

[Ackbach]

It is open since it doesn't contain its limit points. Correct?

What is "it" in this sentence: the original set, or the complement?

 

[Dustinsfl]

What is "it" in this sentence: the original set, or the complement?

Original set.

 

[Ackbach]

Hmm. Well, I'm not sure I agree that the original set is open. Here's an isolated point in the original set: $n=1, m=2$ yields the point $-1/2$. An isolated point is a closed set, because it is a boundary point. So the original set contains some of its boundary points. The question is: does it contain all of its boundary points? Or does it contain all of its limit points?

 

[Dustinsfl]

Hmm. Well, I'm not sure I agree that the original set is open. Here's an isolated point in the original set: $n=1, m=2$ yields the point $-1/2$. An isolated point is a closed set, because it is a boundary point. So the original set contains some of its boundary points. The question is: does it contain all of its boundary points? Or does it contain all of its limit points?

It doesn't contain -1 or 1

 

[Ackbach]

So if it doesn't contain its limit points, but it does contain some of its boundary points, then is the set open, closed, or neither?

 

[Dustinsfl]

So if it doesn't contain its limit points, but it does contain some of its boundary points, then is the set open, closed, or neither?

Neither. So sets that are of the form (1/n) will always contain some boundary points but won't contain their limit point so they will always be neither?