The proof in my book starts out with saying each real number in [0, 1] can be expressed as a binary decimal:
Listing out all the real numbers expressed as binary decimals, one can see that it forms a one to one correspondence with the power set of N since the numbers in binary basically represent the different subsets of N.
however there is a problem with this reasoning. if a binary number ends with an infinite sequence of 0s or 1s then two binary numbers can represent the same real number. therefore it will no longer be a bijection. i can't seem to think of a way to circumvent this problem.
i tried thinking about matching the 2 equivalent binary numbers with the 2 equivalent decimal numbers that all represent the same real number. that way we would get a bijection but it would be from the set [0, 1] with possibly infinitely many numbers repeated to the power set of N. This would no longer be a bijection between just the set of real numbers in [0, 1] and the power set of N so this doesn't seem to work. any help is greatly appreciated. thanks.