My motivation for looking into this was sparked by the 'big-list' question
What are the principal (different) mechanisms of infinite descent proof?
Is the following proof valid?
If an integer $m \gt 0$ is expressed in capital sigma notation,
$$\tag 1 m = \sum_{k=0}^n m_k\, b^k \quad \text{ with } \forall k,\, 0 \le m_k \lt b \text{ and } m_n \ne 0$$
then we say that $m$ has a Base-$b$ representation of degree $n$.
Two Base-$b$ representations of $m$ are said to be identical if they have the same degree and if all the corresponding $m$ coefficients are equal.
Existence
Suppose $a \gt 0$ doesn't have a representation. Using Euclidean division (existence part),
$$\tag 2 a = bq + r \text{ with } 0 \le r \lt b$$
Clearly $a$ must be greater than or equal to $b$, and so $q \gt 0$.
If there was a representation for $q$,
$$\quad q = \sum_{k=0}^n q_k\, b^k$$
then
$$\quad a = r + \sum_{k=0}^n q_k\, b^{k+1}$$
would give a representation of $a$. So $q$ can't have a representation.
Finally, by $\text{(2)}$ we must have $q \lt a$ and by the method of infinite descent we arrive at a contradiction.
So every $a \gt 0$ has a Base-$b$ representation.
Uniqueness
Firstly observe that
$$\tag 3 a = \sum_{k=0}^n x_k\, b^k = x_0 + b \sum_{k=0}^{n-1} x_{k+1}\, b^k$$
Let $a \gt 0$ have two different representations,
$\quad a = \sum_{k=0}^n x_k\, b^k = x_0 + b \sum_{k=0}^{n-1} x_{k+1}\, b^k =$
$\quad\quad\quad \sum_{k=0}^{n'} y_k\, b^k = y_0 + b \sum_{k=0}^{n'-1} y_{k+1}\, b^k$
Clearly $a$ must be greater than or equal to $b$.
Using the above and Euclidean division (uniqueness part with divisor equal to $b$), the integer $c \gt 0$,
$\tag 4 c = \sum_{k=0}^{n-1} x_{k+1}\, b^k = \sum_{k=0}^{n'-1} y_{k+1}\, b^k$
has two different representations.
Clearly we must have $c \lt a$ and by the method of infinite descent we arrive at a contradiction.
So every $a \gt 0$ has a unique Base-$b$ representation.
