countability of limit of a set sequence Let $S_n$ be the set of all binary strings of length $2n$ with equal number of zeros and ones. Is it correct to say $\lim_{n\to\infty} S_n$ is countable? I wanted to use it to solve this problem. My argument is that each of $S_n$s is countable (in fact finite) thus their union would also be countable. Then $\lim_{n\to\infty}S_n$ should also be countable as it is contained in the union.

The collection of all finite strings of $0$'s and $1$'s is countably infinite. The subcollection of all strings that have equal numbers of $0$'s and $1$'s is therefore countably infinite.
I would advise not using the limit notation to denote that collection. The usual notation for this kind of union is  $\displaystyle\bigcup_n S_n$.  
