Infinite intersection of indexed sets

In summary, the greatest element of A_n has n 9s in its decimal expansion and 0.999... is not included in A_n because it is an infinitely repeating decimal that gets closer to 1 but never reaches it.
  • #1
math771
204
0
Consider the set [itex] A_n=\{0.9, 0.99, 0.999,...\} [/itex], where the greatest element of [itex] A_n [/itex] has [itex] n [/itex] 9s in its decimal expansion. Then [itex]0.999\ldots=1\in\bigcap_{n=1}^\infty{A_n}[/itex]. Is this possible even though [itex]\not\exists{n}(1\in{A_n})[/itex]?

Edit: I see that [itex]0.999\ldots=1\not\in\bigcap_{n=1}^\infty{A_n}[/itex]. Sorry :(.
 
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  • #2
It is quite common that the limit of a sequence has some property that no member of the sequence has. Nothing at all strange about that.
 
  • #3
No need to apologize, we all make mistakes sometimes. It's actually a common misconception that 0.999... is equal to 1. In reality, 0.999... is just an infinitely repeating decimal that gets closer and closer to 1, but never actually reaches it. So while it may seem like 0.999... should be in the set A_n, it's actually not included because it's not a finite decimal. Hope that helps clarify things!
 

Related to Infinite intersection of indexed sets

1. What is an infinite intersection of indexed sets?

The infinite intersection of indexed sets refers to the intersection of an uncountable number of sets, where the indexing is done by a set of real numbers or other continuous parameters. In simpler terms, it is the common elements that are present in all of the infinite sets.

2. How is an infinite intersection of indexed sets different from a finite intersection?

An infinite intersection of indexed sets involves an uncountable number of sets, while a finite intersection involves a finite number of sets. This means that the infinite intersection is a more complex and extensive operation, and its result may not always be finite.

3. Can an infinite intersection of indexed sets be empty?

Yes, it is possible for an infinite intersection of indexed sets to be empty. This occurs when there are no common elements present in the infinite sets. However, this is not always the case and depends on the specific sets being intersected.

4. What is the significance of infinite intersections in mathematics?

Infinite intersections of indexed sets play a crucial role in various branches of mathematics, such as analysis, topology, and set theory. They are used to define important concepts such as limits, convergence, continuity, and compactness, among others. They also have applications in fields like physics and computer science.

5. Are there any properties or rules that apply to infinite intersections of indexed sets?

Yes, there are several properties and rules that apply to infinite intersections of indexed sets. These include De Morgan's laws, distributivity, and monotonicity. These properties make it easier to manipulate and solve problems involving infinite intersections.

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