i'll let the map

where B is the set of binary expansions of the real numbers in [0, 1]. Now i know that the set B is in bijection with

since each binary expansion represents some subset of the natural numbers and i will call this function

. my goal is to show that

is a bijection by showing that both g and f are bijections. the issue i have is that 2 elements of B map to one element of [0, 1] so f is not a bijection. if i declare that if an element of [0, 1] maps to 2 elements of B, choose the element of B that ends with infinitely many 0s, then what i have now is a bijection but it doesn't seem to be a bijection between [0, 1] and B since i have removed some number of elements from B. If i call the set A to be the set B with the "equivalent) expansions removed, then i have defined a bijection between [0, 1] and A but i need a bijection between [0, 1] and B. i am a little unsure of how the set B and the function f should be defined.