The problem says "Find a bijecton ."
The hint goes on to say you won't be able to give a specific formula. My main question is "Is this even possible?" I don't see how it is.
Hmm I would do it this way
Map 1 to 3.
Define a bijection between (1,2] and using the usual trick of specifying a point at (2,1) where the number in (1,2] represents an angle in (-pi/2, 0] from the point to the x-axis. (-pi/2 is directly to the left, 0 is straight down).
For non-integers greater than 2, map them to themselves.
For integers n greater than two, map them to n+1.