I've been learning the basics of real analysis lately and have been quite bothered by this question. After going through several proofs, I've been convinced by the proof making use of proper subsets, but I still have a question surrounding the problem left behind, that is, if I try to make a function that maps from the natural numbers $\mathbb N$ to their binary expressions, would it not create a bi-injective function between $\mathbb N$ and $S$?
Notations
The set $S$ is the set of all infinite sequences of "0"s and "1"s is uncountable
$S^n$ is the $n$-th sequence in set S (assuming countable) and $S_m^1$ is the $m$-th element of the first sequence.
Example "proof":
Define$$ f:S\to\mathbb N$$ where $$f(S^n) = \sum_{i=1}^\infty S_i^n * 2^{i-1}$$i.e. $f(1000\dots) = 1, f(0100\dots) = 2, f(0101\dots) = 10$
Since every number have a unique expression in binary, it would map every sequence to a corresponding natural number, making it a bi-injection and $S$ countable.
Now obviously this "proof" is wrong since a similar argument could be applied to $\mathbb R$ which is uncountable, but I don't know what went wrong and how. Could anyone please tell me? Thanks for the help.