# Cardinality: Set of all binary sequence equal c

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. :)
• This is not 1-1, as $011000000\ldots$ has the same image as $01011111\ldots$. You need some extra arguments that the exception points are but countable in number. – Henno Brandsma Feb 28 '16 at 12:48
• Good point: The exceptional points are exactly those that end in an infinite unbroken sequence of 1s; there are countably many of these: the sequence starts at some digit $k$; the previous digit must be a 0; prior digits can be anything, so there are $2^{k-2}$ of these. We end up summing a countable sequence of finite numbers; the result is countable. – John Hughes Feb 28 '16 at 15:29