Explicitness in numeral system 
Prove that for every $a\in\mathbb{N}$ there is one and only one way to express it in the system with base $\mathbb{N}\ni s>1$.

Seems classical, but I don't have any specific argument.

We can express a natural number $a$ to any base $s$ by writing $k = \lceil \log_s a \rceil$, the number of digits we'll need, $a_k=\max(\{n\in \mathbb{N}: ns^k\leq a\}),$ and recursively
$a_{i}=\max(\{n\in \mathbb{N}: ns^i \leq a-\sum_{j=i+1}^k a_j s^j\})$. It's immediate that each of these maxes exists, since $0s^i\leq a$ no matter what and $a s^i\geq a$, and that they're unique by, say, the well-ordering of $\mathbb{N}$.
Then
$$a=\sum_{j=0}^k a_j s^k$$
