Is this collection countable?

Discussion in 'General Math' started by ray_sitf, Jul 15, 2009.

  1. ray_sitf

    ray_sitf Guest

    Let C represent the collection of all unending (infinite) strings of
    zeroes and ones, where both the zeroes and the ones occur an infinite
    number of times. C is evidently unountable.

    Let S belong to C. Let S(n) be an initial segment of S, so that for
    example, S(4) could be 0010 and S(7) could be 0010011.

    Let {S(n)} represent the sum of the digits in S(n), so that in my
    above examples, {S(4)} = 1 and {S(7)} = 3. Of course, {S(n)} just
    counts the number of ones in S(n).

    Let us say that S “converges to zero” if the number of ones in S
    “thins out”. More precisely, if the limit of {S(n)}/n = 0.

    Now let T represent the collection of sequences S that converge to
    zero in this fashion.

    Is T countable?
    ray_sitf, Jul 15, 2009
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  2. ray_sitf

    Virgil Guest

    No. Consider the sequences whose 1's are limited to positions 2^n.

    Each such sequence converges to zero in the sense you defined, but there
    is an obvious bijection between the set of such sequences and the set of
    ALL binary sequences. Just leave out all non-2^n positions in a
    convergent sequence to get a standard sequence.
    Virgil, Jul 15, 2009
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