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Set Thoery Countable Sets

  1. Apr 17, 2008 #1
    1. The problem statement, all variables and given/known data

    Let A be a set of positive real numbers. Assume that there is a bound b such that the sum of any finite subset of A is less than b. Show that A is countable.

    3. The attempt at a solution

    So, my first question is what would be a better approach? Trying to find a function from the natural numbers to A or supposing that A is not countable and deriving a contradiction.

    I'm a little stumped on how to start the construction of such a function.
     
  2. jcsd
  3. Apr 18, 2008 #2

    Dick

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    For all natural numbers n, consider the set of all elements of A that are greater than 1/n, call it A_n. That's a hint. Think about it.
     
  4. Apr 18, 2008 #3
    Well, eventually all the A_n's will capture every element of A, its an increasing sequence.

    I guess I could argue that each A_n is finite since if it werent it proabably would be possible to find some finite subset of A_n who'se sum is greater then b.

    Since A = the union of A_n's then the cardinlaity of A is countable since its the union of countably many countable sets.

    Only problem I have with this argument is I'm not sure how to show with rigor that A_n must be countable or rather (finite).
     
    Last edited: Apr 18, 2008
  5. Apr 18, 2008 #4
    Thanks, I think I got my proof.

    If A_n is infinite then we can find a finite subset of A_n with cardinality m*n where m > b. m,n are natural numbers.

    Thus the sum of the subset must be bounded by b. But each element is greater then 1/n thus the sum is greater then (1/n)m*n = m > b which is a contradiction. Thus A_n must be finite.
     
  6. Apr 18, 2008 #5

    Dick

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    Nice use of the hint.
     
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