Proof

The proof consists of two parts:

Conversion to a-ary representation.

Uniqueness of the representation.

The number m has an a-ary representation. We prove this by induction on m. For m = 0 it is trivial. Now suppose that m > 0 and that the proposition is true for all integers less than m. Let b₀ be the remainder if we divide m by a. Then we have 0 b₀ < a and a | m - b₀.

Since (m - b₀)/a < m there are b₁ , ..., b_k satisfying 0 b_i < a (i=1, ..., k) with

From this it follows that

Uniqueness of the representation:

Again induction on m and again for m = 1 the statement is trivial. For the induction step suppose

and

are two a-ary representations for m. By the assumption on the most significant digit we have b_k 0 and c_l 0. On the one hand, the remainder when m is divided by a is b₀ and on the other hand it is c₀. Hence b₀ = c₀. The number (m - b₀)/a now also has two representations in the a-ary number system:

The induction hypothesis implies k = l and b₁ = c₁ , ..., b_k = c_k.