We define a badly approximable number to be an $\alpha$ such that there exists a constant $c>0$ such that for any $p/q$ we have $|\alpha-p/q|\geq c/q^2$. It is known that a number is badly approximable iff its continued fraction expansion has bounded partial quotients.
Question: give an infinite set of badly approximable numbers, and for a positive real constant $c$ give a bijection between $\mathbb{R}$ and the set $Bad_c:=\{x=[a_0;a_1,a_2,\cdots]\in\mathbb{R}\setminus\mathbb{Q}:sup_{n\ge 0}(a_n)\le c\}$.
An idea for an infinite set of badly approximable numbers would be for instance $[1;0,0,\cdots], [0;1,0,0,\cdots],[0,0,1,0\cdots]$. I think this works, because we have bounded quotients (at most one) and where we simply move the 1 to the next decimal place in each step, of which there will be infinitely many. The second part, constructing a bijection, is much less straightforward. Perhaps we can take $f([a_0;a_1,\cdots])=a_0+a_1+\cdots$; but this is not a bijection, it's only a function... any ideas?