Grandi's series manipulation allowing me to prove 1 = 0?

Question

Let me start by saying I have a very basic background in math, so if I'm using the wrong terms, please feel free to point it out. I know straight away that this is wrong, I'm just curious to know where exactly.

You're probably acquainted with the famous result that $1 - 1 + 1 - 1 + \dots = \frac{1}{2}$. As a quick reminder, this is how that is proven:

1. $S = 1 - 1 + 1 - 1 + \dots$
2. $1 - S = 1 - (1 - 1 + 1 - 1 + \dots)$
3. $1 - S = 1 - 1 + 1 - 1 + 1 - \dots$
4. $1 - S = S$
5. $S = \frac{1}{2}$

But what if we do something slightly different? Something like this, by repeatedly subtracting both sides by 1:

1. $S = 1 - 1 + 1 - 1 + \dots$
2. $1 - S = 1 - (1 - 1 + 1 - 1 + \dots)$
3. $1 - S = 1 - 1 + 1 - 1 + 1 - \dots$
4. $1 - 1 - S = 1 - 1 + 1 - 1 + 1 - \dots$ (Simplifications on the right hand side will just skip to the original form from this point on.)
5. $-S = 1 - 1 + 1 - 1 + 1 - \dots$
6. $1 + S = 1 - 1 + 1 - 1 + 1 - \dots$
7. $1 + S = S$
8. $1 + S - S = S - S$
9. $1 = 0$

I refuse to believe I managed to break math with this kind of thing, so I've been scratching my head trying to find what's wrong with that, but I'm finding it pretty difficult.