Bounds for Integer Expressed in Base k

Theorem
Let $n \in \Z$ be an integer.

Let $k \in \Z$ such that $k \ge 2$.

Let $n$ be expressed in base $k$ notation:
 * $n = \displaystyle \sum_{j \mathop = 1}^s a_j k^j$

where each of the $a_j$ are such that $a_j \in \set {0, 1, \ldots, k - 1}$.

Then:
 * $0 \le n < k^{s + 1}$

Proof
As none of the coefficients $a_j$ in $\displaystyle \sum_{j \mathop = 1}^s a_j k^j$ is (strictly) negative, the summation itself likewise cannot be negative

Thus:
 * $0 \le n$

The equality is satisfied when $a_j = 0$ for all $j$.

We then have: