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- Thread starter dwsmith
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- Jan 26, 2012

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

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

CB

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- Jan 26, 2012

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There can only be two accumulation points. The $(-1)^{n}\in\{-1,1\}$, and the $(1/m)$ are small positive numbers getting ever smaller.If I can't write this an interval, how can I determine if it is open or closed and the accumulation points?

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 ...

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

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- Jan 26, 2012

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What is "it" in this sentence: the original set, or the complement?It is open since it doesn't contain its limit points. Correct?

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

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- Jan 26, 2012

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It doesn't contain -1 or 1someof its boundary points. The question is: does it containallof its boundary points? Or does it contain all of its limit points?

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- Jan 26, 2012

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Neither. So sets that are of the form (1/n) will always contain some boundary points but wont contain their limit point so they will always be neither?