Infinite sequence Xn countability

In summary, the conversation discusses an infinite sequence {xn} where xi is not equal to xj if i is not equal to j. A and B are defined as the sets of all finite and infinite subsequences of {xn}, respectively. The task is to show that A is countable and to show that B is equivalent to the interval (0,1). The first part is achieved by expressing A in terms of Ak and using the fact that the union of countable sets is also countable. The second part can be achieved by using the diagonalization argument.
  • #1
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Homework Statement


{xn} is an infinite sequence and xi ≠ xj if i ≠j. Let A and B denote all finite subsequences of {xn} and all infinite subsequences of {xn}, respectively.

(a) Show that A is countable.
(b) Show that B ≈ (0,1).



Homework Equations





The Attempt at a Solution


We were given a hint to start a like this
(a) Let Ak denote all the finite subsequences using only x1,x2,…xk.
So, each finite subsequence is countable and the union of countable sets is also countable. Therefore, A is countable. He also said we should express A in terms of Ak. I'm not sure how to do that and I'm not sure that what I have is sufficient.
 
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  • #2
(b) We were also given a hint to use the diagonalization argument to show that B ≈ (0,1). I'm not sure how to go about this.Any help would be greatly appreciated!
 

Related to Infinite sequence Xn countability

1. What is an infinite sequence?

An infinite sequence is a list of numbers that goes on forever, without an end point. It is denoted by {Xn}, where n represents the position in the sequence.

2. What is the concept of countability in infinite sequences?

Countability refers to the ability to assign a unique number to each element in a sequence. In the context of infinite sequences, countability means that the elements in the sequence can be enumerated or listed in a specific order.

3. How do you determine if an infinite sequence is countable?

An infinite sequence is countable if each element in the sequence can be assigned a unique number in a specific order. This can be determined by checking if the sequence follows a pattern or if it can be mapped to the natural numbers.

4. What is the significance of countability in mathematics?

Countability is an important concept in mathematics as it helps to classify infinite sets and determine their properties. It also allows for easier analysis and comparison of different infinite sequences.

5. Can there be uncountable infinite sequences?

Yes, there are uncountable infinite sequences, which means that their elements cannot be assigned a unique number in a specific order. An example of this is the set of real numbers, which is uncountable and cannot be mapped to the natural numbers in a one-to-one correspondence.

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