Why is this $0 = 1$ proof wrong?

Question

$0 = 0 + 0 + 0 + ...$
$0 = (1 - 1) + (1 - 1) + (1 - 1) + ...$
$0 = 1 - 1 + 1 - 1 + 1 - 1 + ...$
$0 = 1 + (-1 + 1) + (-1 + 1) + (-1 + ...$
$0 = 1 + 0 + 0 + 0 + ...$
$0 = 1$

I can't really tell what is obviously wrong with this. It seems to use the same logic as we see in the derivation of things like $\sum_{k=1}^{\infty} k = -\frac{1}{12}$ which appears to be a quirky but accepted fact in mathematics.

Answer 1

The third line, what does $$1 - 1 + 1 - 1 + 1 - 1 + \cdots$$

mean? It's certainly not a convergent series like lines one, two, four and five.