Let $b$ denote the radix (=base), suppose that for some $n \in \mathbb Z$ and $m \in \mathbb N$ we know that: $|n|< b^m$ then prove that $$|n|= \sum_{i=0}^{m-1} x_ib ^i$$ Where $x_i \in \{0, 1 ,2 ,3 , 4 \dots b-1 \}$.
Basically this says that if we know that a given number is smaller than some power of the radix, then it must have a representation using the lower powers.
Example: If $|n|< 2^{10}$ then we know this number has some binary representation using at most 9 bits. Give or take a minus sign.
How do we prove this?
I feel this is a job for the Euclidean division algorithm, but not sure how to formulate this. Could I get some help with the steps?