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I recently learned of Cantor's diagonal argument, and was thinking about why there can't be a bijection between any infinite set of integers and any infinite set of real numbers. I understood the basic idea behind the proof, but I was thinking of a particular transformation, for which I don't see why it doesn't form a bijection.
Let's say we want to map all of the integers, to the real numbers between $[-1, 1]$. My transformation is created in such a way, that for every integer, I transform it into a mirrored number that is placed after a decimal.
$$0 \rightarrow 0.0$$ $$1 \rightarrow 0.1$$ $$2 \rightarrow 0.2$$ $$\vdots$$ $$100 \rightarrow 0.001$$ $$\vdots$$
It would seem to me that this is one to one, and every number between $[-1, 1]$ is hit. Even in the example of an irrational number, say $(\sqrt2 - 1)$, there is sum integer of infinite size that maps exactly to $(\sqrt2 - 1)$, because $(\sqrt2 - 1)$ can be written as a decimal made of up an infinite number of integer sitting behind a decimal point. It could be mapped as:
$$...73265312414 \rightarrow 0.41421356237...$$
Now, I'm assuming I was probably not the first person to think of this, so, why does this not work as a bijcetive mapping?