<s>[INST] 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 $s$ ?
 [/INST] Notice that
$$ 1-\frac{1}{2}+\frac{1}{3}-\left( \frac{1}{4}-\dots \right)=1-\frac{1}{2}+\frac{1}{3}-a< 1-\frac{1}{2}+\frac{1}{3}$$
Since $a>0$.
Note:
$$a=\frac{1}{4}-\frac{1}{5}+\dots.$$
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