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.
a_i and a_n are defined the same way, they're just examining two different numbers in the sequence, but each can only be defined after the previous numbers in the sequence have been defined. In your example, after determining a_0=0, you next have to evaluate n=1. This shows a_1 can't be 2, it has to be the largest integer such that 0+a_1/2 <= 0.32, so a_1 has to be 0. Next you evaluate a_2, which given a_0 and a_1 has to be 1.
This process is essentially expressing a number in a different base, but focusing on the non-integer part. What it's asking you to prove is similar to proving that a number in binary will only have digits 0 or 1, and a number in octal will only have digits 0-7.
About your question of definitions more broadly, generally when a~n~ is defined, other subscripts a~i~, a~j~, a~m~, a~xyz~, are defined in the same way, even if that isn't specifically stated.
The a is what carries the properties of the sequence, and any variables in the subscript are just different ways of referencing the index. That can be to communicate a separation between concepts, or to set up relationships like i<=n that will be needed later
Thanks, I was thinking the same. I had already wrote a proof that worked if all a_i are defined like that, but wanted to be sure.