I know a few proofs over this theorem (where $\mathfrak{c}$ is the cardinality of $[0,1]$ and $\aleph_0$ is the cardinality of $\mathbb{N}$) where they construct two injections and then use Schröder-Bernstein (via the Cantor set or something like that).
Now I was wondering if I could do something like this:
Define $s: \{0,1\}^\mathbb{N} \to [0,1]$ by $$s(x) = \sum_{i = 1}^\infty \frac{x(i)}{2^i}.$$
Now this is clearly a surjection because this is just a binary expansion of numbers in $[0,1]$, but not injective because these expansions are not unique. Is there a way to make this work? Can I use this to construct a bijection?