You're right, in the sense that the partial sums are always 0 or 1, so 0.5 away from 0.5. However the mean of the partial sums converges to 0.5 (this is called

You're right, in the sense that the partial sums are always 0 or 1, so 0.5 away from 0.5. However the mean of the partial sums converges to 0.5 (this is called Cesàro summation). It converges "on the average" towards 0.5. However this isn't normally what people mean by convergence.