Proving the Empty Intersection of Intervals using Natural Numbers

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Homework Help Overview

The discussion revolves around proving that the intersection of the intervals (0, 1/n) for n from 0 to infinity is empty. The subject area pertains to set theory and real analysis, particularly focusing on the properties of intervals and limits.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the implications of the intervals (0, 1/n) and how they behave as n approaches infinity. There is an exploration of the conditions under which the intersection would be empty, with some questioning the role of the endpoints and the nature of the intervals.

Discussion Status

Some participants have provided insights into the reasoning behind the proof, suggesting that if the intersection were non-empty, certain conditions would lead to contradictions. There is an acknowledgment of the need for precision in the proof, with suggestions on how to structure the argument effectively.

Contextual Notes

Participants note that the original problem involves intervals that do not include 0, and there is a contrast made with a modified version of the problem where the intervals would include 0, leading to a different outcome. This highlights the importance of definitions in the proof.

cragar
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Homework Statement


Prove that \bigcap_{n=0}^{\inf} (0,\frac{1}{n})=\emptyset

The Attempt at a Solution


since 0 is not included in our interval. eventually I will get to
(0,0) because I could pick a real as close to zero as I wanted and there would be a natural such that \frac{1}{n}<y therefore this intersection is empty.
but if my orginal intersection was [0,1/n] then this would not be empty.
 
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You have the right idea, and you're correct that if (0,1/n) was replaced by [0,1/n], the intersection would not be empty. It would contain exactly one point: 0.

You could word your proof a bit precisely as follows:

Suppose the intersection is non-empty. Then there exists some x in the intersection:

x \in \bigcap_{n=1}^{\infty} (0, 1/n)

Since x is in the intersection, it means that x must be in every interval of the form (0, 1/n), and this is impossible because...
 
ok thanks for the input
 
you might want to show that if x > 0, there exists k in N with 1/k < x, first.
 

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