Bijective function- for reals on set [0,0.1)

Question

Find a bijective function from the reals to the set [0, 0.1)

I think Cantor has something to do with it but I am unsure? Thoughts?

Answer 1

Hint: Start by finding an explicit bijection $f:\Bbb R\to(0,1).$ We will then adjust that to a bijection $g:\Bbb R\to[0,1)$ by letting $g(0)=0,$ $g(n)=f(n-1)$ for positive integers $n,$ and $g(x)=f(x)$ whenever $x$ isn't a nonnegative integer. Once you've proved that $g$ is such a bijection, then you can compose $g$ with an easy bijection $[0,1)\to[0,0.1).$

Alternately, you can start with injections $f:[0,0.1)\to\Bbb R$ and $g:\Bbb R\to[0,0.1),$ then "crank the handle" of the Cantor-Bernstein Theorem, as described in this fine answer.