I try to show that the Baire space $\Bbb N^{\Bbb N}$, with regular product metric, is homeomorphic to the unit interval of irrationals $(0,1)\setminus\Bbb Q$. I already know that the needed function is using continued fractions
$$a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3+\cfrac1{a_4+\cfrac1\ddots }}}}$$
My question is how to show that this function actually fulfills what we want? - how to show it can only represent irrationals by this continued fraction? " every irrational in the unit interval? " it is 1-1?