I am having trouble understanding the problem. What exactly is your question?
Prove that the Odometer Principle with base gives the representation for the natural number .
Odometer Principle to find the successor of a natural number to base : Start by considering the rightmost digit.
- If the digit we are considering is not , then replace it with the next digit in order, and terminate the algorithm.
- If we are considering a blank space (to the left of all digits) then write in it the digit , and terminate the algorithm.
- If neither of the above holds then we are considering the digit . Replace it with the digit , move one place to the left, and return to the first bullet point.
Proof. We use induction on . For , . Now suppose that for , . So now we look at (the rightmost digit) and apply the Odometer Principle?