I was working on this problem for class
B is the set of all infinite binary strings. Show B is uncountable.
The given answer uses a diagonal argument similar to the Cantor proof, however that wasn't my first thought.
I first thought to do the following:
Index every digit in a binary string $1,2,3,…$. The number of digits in a particular string is $\mathbb{N}$, and since each digit can earlier be a $0,1$ we have that our set contains
$$
2^{|\mathbb{N}|}
$$
Strings, which is the same as the cardinality of the continuum. Therefore, B is uncountable.
Is this sufficient? I'm not sure how infinite cardinalities hold up to operations like that.
Best Answer
If you already know that $2^{\Bbb N}$ is uncountable, then what you have there is enough. As you have shown that there is an injection $2^{\Bbb N}\to B$, which implies $$ \left\lvert2^{\Bbb N}\right\rvert\leq |B| $$