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Math Help - Proof that the intersection of infinitely many countable sets is countable.

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    Proof that the intersection of infinitely many countable sets is countable.



    Did I prove this sufficiently?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by davismj View Post


    Did I prove this sufficiently?
    You mean union.

    a) Proving by induction implies that the union of a fixed arbitrarily large number of countable sets is countable. It says nothing about letting n range freely over the naturals.

    b) The second part confuses me. The best way to do this is to first prove that if f:\mathbb{N}\mapsto E is surjective, then E is countable and then note that g:\mathbb{N}^2\mapsto\bigcup_{j=1}^{\infty}A_n where g(m,n)=f_m(n) (where f_m is the bijection from \mathbb{N} to A_m) is a surjection.
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