Am I meant to assume a_i is defined the same way as a_n for each of 1<= i <= n-1 ?
If so, I think I see the proof by induction on n, but the question just says a_i is defined for each 1<=i<=n-1, not that it is defined in that way. Is the question just overly vague or am I missing something obvious?
If only a_n is defined as the greatest integer such that the sum from 0 to n of each a_i/k^i is <=x, then I think there are counterexamples to the hypothesis, right?
Like if x=0.32, k=2, n=2, then a_0=0, and the inequality is satisfied by a_1 = 2 > k-1 = 1 and a_2 = -3 < 0.
Edit: read my follow up comments
~~There is information missing, so you should ask your professor or move on to a different question.~~
~~It might be that it means to say a_0<=a_1<=…<=a_n integers have been defined~~
~~But that’s just a guess I haven’t checked the validity of.~~
Looking at it again I think you should read the question inductively.
So a_1 is defined as the largest integer such that the inequality for n=1 holds, then given that a_1, a_2 is the largest integer such that the inequality for n=2 holds etc
So if x=0.32 and k=2 then a1=0, a2=1, a3=0 etc
I’ll use this final comment to complain about how lazy some mathematics professors are with formulating their questions that it sometimes takes more time to decipher the question than to come up with the correct answer.