According to the last step in proof of the unmeasurability of Vitali_set, it said that summing infinitely many copies of the constant $\lambda(V)$ yields either zero or infinity, according to whether the constant is zero or positive. It sounds pleasing to my ear, but I still have a bit doubt in the reason of the sum of infinitely many copies of a positive real constant would definitely yield infinite.
Actually, $< \sum_{i=0}^n c>_{n=0}^{\infty}$ is indeed an strict increasing series when $c>0$. However, this fact seems cannot guarantee the inevitability of $\sum_{i=0}^\infty c=\infty$.
Take an example in infinite product. $1,2,4,8,16...$ is actually a strict increasing series too, but $\prod_{i=0}^{\infty}2$ can yield $0$ in some cases.
Moreover $9,99,999,\ldots$ is also a strict increasing series, but in some theory $...999$ is not a infinite but $-1$.
So my question on what basis can the conclusion that $\sum_{i=0}^\infty \lambda(V)$ is necessarily not between $1$ and $3$ be concluded?