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 N\mathbb{N}, we can define a sequence (an)(a_n) 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 ana_n as follows:

an=nmodka_n = n \mod k where kk is a large enough integer depending on nn, ensuring that each natural number appears infinitely often in the sequence.

Alternatively, we can explicitly define the sequence as: an=n2ka_n = \left\lfloor \frac{n}{2^k} \right\rfloor 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 nn grows.
  • Each natural number will appear repeatedly in the sequence.
  • As nn \to \infty, the limit points of the sequence will be all the natural numbers N\mathbb{N}, because the sequence visits every natural number arbitrarily close for large enough nn.

Thus, the set of limit points of the sequence is N\mathbb{N}.


Would you like more details or have any questions? Here are some related questions to think about:

  1. How does the density of the sequence affect its limit points?
  2. Can you create a sequence whose set of limit points is only a finite subset of natural numbers?
  3. What would the limit points be if we modified the sequence to include non-natural numbers?
  4. Can we define a sequence that has limit points both in N\mathbb{N} and outside of it?
  5. 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