U just took the wrong thing and proved it wrong.....it never said z+z=z.....it said that 1-1+1-1+1-1+1-1.... infinity can either be 0 or 1 which is equiprobable which means sum will be either 0 for 1....thus using Bayes theorem it's 1/2..... It's the assumption that infinity can be made finite and can be sorted into acting like a finite number which is itself wrong