Proving an intersection empty set

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SUMMARY

The intersection of the open intervals \(\bigcap^{\infty}_{n=1}(0,1/n)\) is proven to be the empty set. The Nested Interval Property indicates that for closed intervals, the intersection is non-empty, but this does not apply to open intervals. To demonstrate that any \(x\) in the interval (0,1) does not belong to (0,1/n), one must find an \(n\) such that \(x \geq 1/n\). This effectively shows that as \(n\) increases, the intervals shrink, confirming the intersection is indeed empty.

PREREQUISITES
  • Understanding of open and closed intervals in real analysis
  • Familiarity with the Nested Interval Property
  • Knowledge of supremum and infimum concepts in set theory
  • Basic proof techniques in mathematical analysis
NEXT STEPS
  • Study the properties of open and closed intervals in real analysis
  • Learn about the Nested Interval Theorem and its implications
  • Explore the concept of limits and convergence in sequences
  • Investigate the relationship between supremum and intersection of sets
USEFUL FOR

Mathematics students, particularly those studying real analysis, educators teaching interval properties, and anyone interested in advanced proof techniques in set theory.

rbzima
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I'm trying to prove

[tex]\bigcap^{\infty}_{n=1}(0,1/n) =[/tex] EMPTY SET.

One thing that I can't seem to bypass is getting past the closed intervals as stated in the Nested Interval Property, which states "For each [tex]n\in[/tex] N, assume we are given a closed interval [tex]I_n = [a_n, b_n] = {x \in[/tex] R [tex]: a_n \leq x \leq b_n}[/tex]. Assume also that each [tex]I_n[/tex] contains [tex]I_n_+_1[/tex].

Therefore, [tex]\bigcap^{\infty}_{n=1}I_n \neq[/tex] EMPTY SET.

Basically, I'm stuck at taking care of the open interval part of this proof. The original for R was proved by showing sup A = x, then x is between the closed interval. Any advice would be great!
 
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Obviously the intersection is a subset of (0,1), Why? So it will suffice to show that for any x in (0,1) that there exists a set of the form (0, 1/n) to which x does not belong. So basically given x in (0,1) can you figure out how to find n such that x does not belong to (0, 1/n)?
 

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