I accept that two numbers can have the same supremum depending on how you generate a decimal representation. So $2.4999\ldots = 2.5$ etc.

Can anyone point me to resources that would explain what the below argument that shows $999\ldots = -1$ is about?

Here is the most usual proof I see that $0.999\ldots = 1$:

> 
> $x=0.999\ldots$
> 
> 
> $10x=9.999\ldots$
> 
> 
> $10x - x = 9$
> 
> 
> $x=1$
> 
> 
> 

Using this same argument template I can show $999\ldots=-1$:

> 
> $x= \ldots9999.0 $
> 
> 
> $0.1x= \ldots9999.9$
> 
> 
> $0.1x - x = 0.9$
> 
> 
> $x=-1$
> 
> 
> 

What might this mean?

Edit from one of the comments:

$$\sum\_{k=0}^{\infty}{9 \cdot 10^k}=-1$$