Use induction to prove that radix sort works. Where does your proof need the assumption that the intermediate sort is stable?

We can take the following invariant:

At the beginning of the

forloop, the array is sorted on the last $i - 1$ digits.

**Initialization**. The array is trivially sorted on the last 0 digits.

**Maintenance**. Let's assume that the array is sorted on the last $i - 1$
digits. After we sort on the $i$th digit, the array will be sorted on the last
$i$ digits. It is obvious that elements with different digit in the $i$th
position are ordered accordingly; in the case of the same $i$th digit, we still
get a correct order, because we're using a stable sort and the elements were
already sorted on the last $i - 1$ digits.

**Termination**. The loop terminates when $i = d + 1$. Since the invariant
holds, we have the numbers sorted on $d$ digits.

We use the assumption in the maintenance explanation.