Let $n_1 > 0$ be a multiple of $9$. Suppose that we add up all the (base-10 digits) of $n_1$; denote this sum by $n_2$. Then add up all the digits of $n_2$ to get $n_3$, and all the digits of $n_3$ to get $n_4$, and so on. This produces a sequence of numbers $n_1, n_2, n_3, \dots , n_k,\dots$
Use induction to prove that $n_k = 9$ for all large enough $k$. When you write up your solution, clearly state your induction hypothesis.
So far, I have tried to write down the statements above in mathematical form but I am encountering a few problems. By adding up all the multiples of $9$, wouldn't I go to infinity as there are an infinite number of multiples of $9$. Also, how would I develop a base case and induction hypotheses when relating to sums?