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 show existence by (strong) induction. Suppose the result is true for all $m\lt n$. Show it is true at $n$.
Let $s^k$ be the largest power of $s$ that is $\le n$. Then $1\le n/s^k \lt s$. So if $d=\lfloor n/s^k\rfloor$ then $s=ds^k +m$ where $0\le m\lt n$ and $m\lt s^k$. If $m=0$ we are finished, else apply induction hypothesis.
For uniqueness, again we can use induction. The lead digit is uniquely determined by the condition in the preceding paragraph. 
Another way: Uniqueness is particularly easy to prove by an induction running in the other direction. Take a particular representation, of a possible several. The "units" digit $d_0$ is uniquely determined as the remainder when $n$ is divided by $s$. But then the next digit $d_1$ is the remainder when $(n-d_0)/s$ is divided by $s$. This is determined completely by $n$. But then the next digit is the remainder when $(n-d_0-d_1s)/s^2$ is divided by $s$, again completely determined by $n$. And so on. 
Existence can be proved in the same way. Let $d_0$ be the remainder when $n$ is divided by $s$. Then $n=ms+d_0$, and by induction hypothesis $m$ has a representation. 
