Math Problem Statement
Construct a sequence whose limit point is set of natural number
Solution
To construct a sequence whose set of limit points is the set of natural numbers , we can define a sequence that approaches every natural number infinitely many times. One way to construct such a sequence is to define a sequence that cycles through the natural numbers but gets closer and closer to each one.
Example sequence:
Define the sequence as follows:
where is a large enough integer depending on , ensuring that each natural number appears infinitely often in the sequence.
Alternatively, we can explicitly define the sequence as: This method ensures that each natural number is repeatedly visited and thus becomes a limit point.
Explanation:
- The sequence starts by cycling through smaller integers and revisits them infinitely often as grows.
- Each natural number will appear repeatedly in the sequence.
- As , the limit points of the sequence will be all the natural numbers , because the sequence visits every natural number arbitrarily close for large enough .
Thus, the set of limit points of the sequence is .
Would you like more details or have any questions? Here are some related questions to think about:
- How does the density of the sequence affect its limit points?
- Can you create a sequence whose set of limit points is only a finite subset of natural numbers?
- What would the limit points be if we modified the sequence to include non-natural numbers?
- Can we define a sequence that has limit points both in and outside of it?
- How would this approach change if we wanted the limit points to be integers, instead of just natural numbers?
Tip: Limit points of a sequence are the values that the sequence gets arbitrarily close to infinitely often.
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Math Problem Analysis
Mathematical Concepts
Sequence and Series
Limit Points
Set Theory
Formulas
a_n = n mod k
a_n = ⌊n / 2^k⌋
Theorems
Limit Point Theorem
Suitable Grade Level
Undergraduate Mathematics
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