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Every countably infinite set has a countably infinite set

  1. Mar 1, 2012 #1
    1. The problem statement, all variables and given/known data
    Show that every infinite set contains a countably infinite set.

    2. Relevant equations

    3. The attempt at a solution
    Let A be an infinite set

    Since A is infinite and not empty, there exists an a∈A
    Let f be a mapping such that for all n in ℕ such that f(n)=[itex]a_{n}[/itex] for some [itex]a_{n}[/itex].

    Then f(1)=[itex]a_{1}[/itex]. A is an infinite set, so there exists another element [itex]a_{2}[/itex] that is not [itex]a_{1}[/itex] such that f(2)=[itex]a_{2}[/itex] and so on.

    This makes f a 1-1 function.

    Since every element of ℕhas an image, the subset generated by f is onto ℕ.

    Therefore, since f creates a subset of A, infinite subsets have a countable subset.

    Is this a valid proof on why infinite sets have a countably infinite subset?
  2. jcsd
  3. Mar 1, 2012 #2


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    Science Advisor
    Homework Helper

    Yes, it is. I would rephrases some of the sentences, but in spirit it's correct.
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