As is well known, we can expand every positive integer $n$ to a base $b \in \Bbb N$ in the form $$n = \sum_i a_ib^i ,\ \ \ 0\leq a_i \leq b_i-1$$ uniquely.
Less well known is that we can do this for $b= c/d \in \Bbb Q^{+} \text{ with } c > d$ in the form $$n = \sum_i a_i b^i,\ \ \ 0\leq a_i < c ,\ \ \ \ c,d,a_i,n \in \Bbb N$$This expansion will not be unique in general and the proof I know of is by induction. One simply looks at the maximum value reachable by using upto $k$ terms and shows that the $b^{k-1}$ term is less than this value.
Is there a more intuitive proof that avoids computation of these results? Preferably, whatever the strategy is would prove both results at once and show uniqueness in the case of integers too.