For a positive integer denote by $f(n)$ the largest integer $k$ with $2^k | n$. Let $a_0=0$ and consider the sequence given by $\frac{1}{a_n}=1+2f(n)-a_{n-1}$. Prove that this sequence contains every positive rational number exactly once.
I started with the following:
By the inductive hypothesis, assume that there is some set $S$ with the elements $\{a_0, a_1,..., a_n|a\in\mathbb{Q^+}\}$ such that no $a_i = a_j$ for distinct $i,j $ $(0\leq i,j\leq n)$.
Now we need to prove $a_{n+1}$ is a positive rational and not equal to any other element in $S$. At this point I'm having some troubles. It's easy to see that $a_{n+1}$ will always be a rational, but how do you prove that it will always be positive with the $f(n)$. And for the second part of the inductive step, I tried a proof by contradiction, assuming there was some $a_k = a_{n+1}$. I kept hitting dead ends at this point.
You prove that $a_{2n+1}>a_{2n}$, which eventually leads to proving $a_{2n}<1$ and $a_{2n+1}>1$. This ultimately solves the positive rationals issue and leads to showing no repetitions exist within the set $S$.