How to Prove the Empty Intersection of Infinite Intervals?

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SUMMARY

The discussion centers on proving that the infinite intersection of the intervals En = [n, ∞) is empty within the context of the real numbers R. The key insight is that for any real number x, there exists a positive integer n such that x < n. This fact directly leads to the conclusion that no real number can belong to all intervals En, thus confirming that the infinite intersection is indeed empty.

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Homework Statement


If X = R with a special metirc, En = [n, infinity) ()
How can I show infinite intersection of En is empty?

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The Attempt at a Solution

 
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I am not sure what you mean by "special metric", since the statement you are trying to show has nothing to do with a metric.

To show that the intersection is empty use the fact that for any real number x there exists a positive integer n such that x<n.
 

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