Zero Padded Basis Representation/Informal Proof

Theorem

Let $b \in \Z: b > 1$.

Let $m \in \Z_{> 0}$.

For every $n \in \Z_{\ge 0}$ such that $n < b^m$, there exists one and only one sequence $\sequence {r_j}_{0 \mathop \le j \mathop \le m - 1}$ such that:

$(1): \quad \ds n = \sum_{j \mathop = 0}^{m - 1} r_j b^j$

$(2): \quad \forall j \in \closedint 0 {m - 1}: r_j \in \N_b$

Informal Proof

The sequence $\sequence {r_j}_{0 \mathop \le j \mathop \le m - 1}$ is the basis representation from the Basis Representation Theorem padded with $0$ to the length of $m$.

$\blacksquare$