Lim sup lim inf of rationals in [0,1]

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SUMMARY

The limit superior (lim sup) and limit inferior (lim inf) of the set of all rational numbers in the closed interval [0,1] are both equal to 1. To prove this, one must demonstrate that 1 is an upper bound for the set of rationals in [0,1] and that no number less than 1 serves as an upper bound. The discussion emphasizes the importance of understanding limit points within this context to validate the conclusions drawn.

PREREQUISITES
  • Understanding of limit superior and limit inferior concepts in real analysis
  • Familiarity with the properties of rational numbers
  • Knowledge of upper bounds and limit points
  • Basic proficiency in mathematical proof techniques
NEXT STEPS
  • Study the definitions and properties of limit superior and limit inferior in real analysis
  • Explore the concept of limit points in metric spaces
  • Investigate the completeness of the real numbers and its implications for rational numbers
  • Practice proving upper bounds and limit points with various sets of numbers
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Mathematics students, educators, and researchers interested in real analysis, particularly those focusing on the properties of rational numbers and limits.

dopeyranger
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what is the lim sup of the set containing all rationals in the closed interval [0,1] ?
and what is the lim inf?

How do I prove that the value is correct?
 
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dopeyranger said:
what is the lim sup of the set containing all rationals in the closed interval [0,1] ?
and what is the lim inf?

How do I prove that the value is correct?

Well, if you think the lim sup might be, for example, 2 [mind you, I'm not saying it is], you would have to show two things:

1. 2 is an upper bound for your set of numbers
2. No number less than 2 is an upper bound for your set of numbers.

In other words, you use the definition. What is your opinion in regard to 2 being the sup? And why? That might get you thinking about what you need to do.
 
Have you found the limit points of the set of rationals in [0,1]?
 

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