A basic doubt on a infinite series problem I see in Rudin the following statement is claimed for the following convergent series: 
$$1-\frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \dots$$
If $s$ is the sum of this series then $$s \lt 1 -\frac{1}{2} + \frac{1}{3}$$. How is that possible to tell without knowing $s$ ?

$$S_{2k+1} = 1-\frac{1}{2} + \frac{1}{3} - \left(\frac{1}{4} - \frac{1}{5}\right) - \ldots -\left(\frac{1}{2k} - \frac{1}{2k+1}\right)  < 1-\frac{1}{2} + \frac{1}{3}-\left(\frac{1}{4} - \frac{1}{5}\right)$$ and $S_{2k+1}\to s$ imply
$$s \leqslant 1-\frac{1}{2} + \frac{1}{3}-\left(\frac{1}{4} - \frac{1}{5}\right) < 1-\frac{1}{2} + \frac{1}{3}.$$
