How do I prove the cardinality of the set of all binary sequences equal c?
I know I have to find a bijective function from (0,1) to the set of all binary sequences. But I can't think of one.
Cantor's diagonal argument only shows it is uncountable, i.e. cardinality greater than d.
How about $$\{b_i\} \mapsto \sum_i b_i 2^{-i},$$i.e., treat each binary sequence like $01100000...$ as a binary fraction, i.e., $.011 = 3/8$. The non-finite ones will be where the irrationals get produced. :)