So there is this extra exercise in my textbook. I looked it over with the TA and neither of us could solve it. The exercise goes like this:
Let $\alpha$ be positive and rational. Then, choose the smallest natural number $N_0$ such that $1/N_0 \leq \alpha$. Now let $\alpha_1=\alpha-1/N_0$. Now choose $N_1$ as the smallest natural number such that $1/N_1 \leq \alpha_1$ etc.
The exercise is to prove that this algorithm terminates for every rational $\alpha$. (i.e. at some point $\alpha_i=N_i$ and it will end as $0$) It's easy to prove that $\alpha_i$ becomes arbitrarily small, but I don't see any approach to prove it reaches zero.