Question regarding a proof of the uncountability of real numbers I recently read theorem $2.14$ from baby Rudin. Here is what it states:
Let $A$ be the set of all sequences whose elements are the digits $0$
and $1$. This set $A$ is uncountable.
He goes on to prove this theorem and then continues with the following comment:
Readers who are familiar with the binary representation of the real
numbers (base 2 instead of 10) will notice that Theorem $2.14$ implies that the
set of all real numbers is uncountable. I read about the binary representation of the real numbers here on stack exchange but I'm wondering why this isn't implied with the decimal representation of the real numbers.
I construct a map $f$ which takes an element of $A$ and maps it to the number $1,s$. So if for example $s\in A $ is $01011010...$ then the number it corresponds to is $1,01011010...$
Then the set $f(A)$ is uncountable and since $f(A)\subset \mathbb{R}$, $\mathbb{R}$ is uncountable as well. Is this also a valid proof or is there something wrong with it?
 A: Each argument has an advantage, both can be made correct. A common mistake in Rudin's scenario is to identify each sequence with a binary number in $[0,1]$. This looks naively like a bijection (it is surjective) but it fails to be injective, because (for example) the following two sequences represent the same number:
$$ 0,1,1,1,1,1,1,\ldots \quad\text{vs}\quad 1,0,0,0,0,0,\ldots$$
As this answer on MO notes, there are only countably many exceptions which prevent this from being injective, and hence $[0,1]$ is uncountable.
Meanwhile, your argument is actually more direct - it is obviously injective. This is because in decimal, the ambiguity above is between (say) $0.9999\ldots$ and $1.000\ldots$, and your image only ever contains one of them. However, your function is very far from surjective, so it leaves open the possibility that there are many more decimal numbers than there are 0-1 sequences (and there are not).
Of course, in your example, starting with a 1 means that your image is a subset of $[1,2]$. But that hardly matters.
