This seems to be the main point. You can identify all of the points.
Every real number in [0,1] corresponds to an infinite binary fraction.
(The slight complication is that some numbers have two representations,
such as 1.0 and 0.1111..., the analog of 0.9999... in decimal. But this
can be dealt with since there are only countably many such numbers.)
Again, all that is needed is to agree that there is a one to one
correspondence between the real numbers and the infinite binary fractions
(modulo the point above). This applies to all the reals, even the
ones which are transcendental, random, or what have you. Each of them
corresponds to an infinite binary fraction.
Once you agree on this, it immediately follows that C = 2^aleph-null.
Hal