User blog:PsiCubed2/An FS-independent injection from the Countable Ordinals to the reals? (help me find the catch)

Inspired by this post by Edwin Shade, I've got the following idea for encoding ordinals with real numbers between 0 and 1:

(1) The finite natural number n will be coded as the decimal expansion of 1/(n+1).

(2) Given a real number x, we break it into a sequence of real numbers x1, x2, x3, ... using the following ω²-level ordering:

1, 2, 4, 7, 11, ....

3, 5, 8, 12, 17, ...

6, 9, 13, 18, 24, ...

10, 14, 19, 25, 32, ...

15, 20, 26, 33, 41, ...

IOW, the to get the digits of x1 we take the 1st, the 2nd, the 4th, the 7th, the 11th... digits of x, and so on.

The ordinal represented by x is defined as the supermum of the ordinals represented by x1,x2,x3.

And that's it.

Now, it seems that I've just created an FS-independent injection from all the countable ordinals to the reals... and this just seems too good to be true. So what's the catch?