If the proof were just that the sum converges to some number 0 <= x <= 1, then you'd be right. But it's showing specifically that the sum lies between the firs

If the proof were just that the sum converges to some number 0 <= x <= 1, then you'd be right. But it's showing specifically that the sum lies between the first term s_0 (which is strictly less than 1) and the sum of the first two terms s_0 - s_1 (which is strictly greater than 0). So there's no possibility for it to converge to a value outside of the interval [s_0-s_1,s_0], which is itself strictly contained inside [0,1].