G-adic numbers/Convergence/Exercise

Let ${}g\in \mathbb {N}$ , ${}g\geq 2$. A sequence of digits, given by

$z_{i}\in \{0,1,\ldots ,g-1\}{\text{ for }}i\in \mathbb {Z} ,\,i\leq k,$

(where ${}k\in \mathbb {N}$) defines a real series

$\sum _{i=k}^{-\infty }z_{i}g^{i}.$

Prove that such a series converges to a unique non-negative real number.

Create a solution