While trying to explain to someone else how you can have a bijection between $[0,1]$ and $[0,1]\times[0,1]$, I found an issue in the usual bijection that we use.
The usual bijection that I'm talking about is the one where $x=0.a_1a_2a_3a_4\ldots$ is sent on $(0.a_1a_3\ldots, 0.a_2a_4\ldots)$. First, I was wondering where is sent the real $1$. Then, I thought that since $1=0.999\ldots$, it is sent on $(0.999\ldots, 0.999\ldots)=(1,1)$.
However, this double-writing of a real number with a finite number of nonzero decimals leads to a non-injective map.
For example, the image of $0.1$ is $(0.1,0)$ whereas the image of $0.00\overline{90}$ is $(0.0\overline9,0)=(0.1,0)$ so the map is definitely not injective.
Is there something wrong with my comprehension of the bijection or there is an issue that people usually don't cover when talking about that bijection?
Note that I'm not doubting in any way that there is a bijection between $[0,1]$ and $[0,1]\times[0,1]$, I'm just wondering if the usual one is indeed a bijection.
