My students found an old problem given in my school in 2007 (probably from a Honor Calculus class) and had been trying to solve for some time. Here is the problem:
Prove or disprove: there exists a bijection $a$ from $\mathbb N$ onto $\mathbb Q$ such that $\sum_{n=1}^\infty (a_n-a_{n+1})^2$ is convergent. (With $a_n=a(n)$)
I have to confess that I am clueless on the method to deal with this problem. My intuition is that the sum represents the square of the distance traveled when visiting all rational numbers, but there is so many rationals that the sum should be infinite. On the other hand, the density of Q means that I can travel the distance between consecutive rationals of my travel can be arbitrarily small, so I am confused...
The only thing we could prove is that $\sum(a_n-a_{n+1})$ is divergent (otherwise, $\{a_n\}$ would be bounded, a contradiction).
Any clue is welcome.