Prove that there is a bijection between (0,1) and (b,infinity) for any real number b.
A function is needed. I've though about the function f(x)=b/x but it won't work if b<=0. Any idea?
$\displaystyle f (x)= tan(\pi x - \pi/2)$ is (I believe) standard for expanding $\displaystyle (0, 1)$ to $\displaystyle \mathbb{R}$, so I would use this.
This can be relatively easily manipulated to give you the function you want.
(Half it to give you a bijection between $\displaystyle (0, 1)$ and $\displaystyle (0, \infty)$, then scale).