Given a two element set $\{0,1 \}$, we want to find a bijection between $\{ 0,1 \}^{\omega}$ and a proper subset of itself

Question

Let $X = \{ 0,1 \}$. Find a bijective correspondence between $X^{\omega} = X \times X \times .... $ and a proper subset of itself.

Attempt to the solution:

Notice that since $\{0 \} \subset \{0,1 \}$, then the set $B = \{0 \} \times \{ 0 \} \times .... $ is a proper subset of $X^{\omega}$. So we may define a function $f: B \to X^{\omega} $ as

$$ f(0,0,0,0,.....) = (1,1,....) $$

and this would trivially be bijective. Is this correct?

Answer 1

Anything like @J.W. Tanner suggests $f_n(a_1,a_2,\dots)=(0,\dots,0,a_1,a_2,a_3, \dots)$, where the first $n$ coordinates are zero, will work.