You can assume LHS=RHS in the first step... there's just the matter of proving it, which is why he uses this proof. However, the prove does make assumptions, making it untrue.
Because LHS=0, the last number in series RHS must be -1. The second step assumes that the last number in series RHS is 1, but it isn't.
0=1-1+1-1+1-1+1-1+1-1+1-1+1-1+...-1
RHS=1-(1-1)-(1-1)-(1-1)-(1-1)-(1-1)-(1-1)-...1
RHS=1-0-0-0-0-0-0-...1
RHS=0
But we started with LHS=RHS.
Therefore LHS=0.
Therefore 0=0. Who knew? I really thought 0=42.
What's the point of this? Are you trying to trick us?
What's the square root of 1000?