Label the set of all binary series with an infinite amount of 0's and 1's as $C$.
It's easy to prove that the set (labeled $A$) of all binary series with a finite number of 1's is countable. I can then prove in an almost identical way that the set (labeled $B$) of binary series with a finite number of 0's is countable.
The set (labeled $D$) of all binary series is a direct sum: $A+B+C=D$.
$D$'s cardinality is a known $\mathfrak c$ (continum), so we get:
$$|A|+|B|+|C| = |D|\\ \aleph_0+\aleph_0 + |C|=\mathfrak c\implies |C| =\mathfrak c$$
What do you think of the proof? Thanks for your time.