Is this collection countable?

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?

 

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.