I am trying to prove that $X^{\omega}$ where $X=\{0,1\}$ (i.e, the set of all sequences of $\{0,1\}$) is bijective to the collection of all countable subsets of $X^{\omega}$. I can write an injection from $X^{\omega}$ to a countable subset of itself, by the following method: If $\{x_1,x_2 \cdots\} \in X^{\omega}$, then my corresponding subset will be $\{ (x_1,0,0,0 \cdots), (0,x_2,0, \cdots).... \}$ (This function basically captures where all the 1s occur, so that this is an injection is shown). The direction from the set of all countable subsets of $X^{\omega}$ to $X^{\omega}$ seems tricki…