We may generalize the decimal expansion of a number by replacing the integer 10 with any integer . If , let denote the greatest integer greater than . Assuming the integers have been defined, let , denote the largest integer such that

Show that for each .

*Proof.* Suppose otherwise, that for some we have . Hence,

This contradicts that is the greatest integer such that

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Very nice and conscious proof. But I think you have to proove also .

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for that, let n>=1 and note that a_n-1 is the greatest integer such that the sum from k=0 to k=n-1 of a_k/b^k <= x, but this sum + 0/b^k is also less than or equal to x hence a_n must be at least 0