(3) Let $X := (0,1)$ be the open interval between 0 and 1, and let
$Y:= (0,1) \times (0,1)$ be the open square in the plane. We show $X$ is
bijective to $Y$ in this problem, and along the way illustrate important
notions in the proof of Cantor-Schröder-Bernstein theorem.
In the following, we represent any number in the set $X$ by a bi-
nary expansion, and make the convention that we do not allow a
finite binary expansion; for example, 0.001 must always be replaced by
0.000111111..., etc.
(a) Show that the map $f: X \rightarrow Y$ defined by
$$f: 0.a_1a_2a_3 \cdots \mapsto (0.01111111\cdots, 0.a_1a_2a_3\cdots)$$
is injective.
(5 points)
(b) Define $g: (0, 1) \times (0, 1) \rightarrow (0,1)$ by
$$g: (0.a_1a_2a_3\cdots, 0.b_1b_2b_3\cdots) \mapsto 0.a_1b_1a_2b_2a_3b_3\cdots \in (0,1).$$
Show that $g$ is injective.
(10 points)
Remark. As a consequence, we know by Cantor-Schröder-Bernstein
theorem that $X$ is bijective to $Y$.
