DedekindМожно нарисовать картинку из которой будет видно, что в точности происходит. В частности, справедливость утверждения будет наглядно видна.
Во-первых, немного изменим обозначения. Будем обозначать наши множества

и

(вначале вы так и пишете, но потом почему-то меняете обозначения на

и

):

и

- инъекции. Вместо обозначения

будем использовать

, как в ссылке на википедию:

(обозначения

у нас будут зарезервированы для других множеств)
Представим себе, что множества

и

с соответственными отображениями - это два зеркала друг напротив друга, и начнем процесс "отзеркаливания", то есть индукционно строим последовательность упорядоченных пар

следующим образом:

Тривиально по индукции доказывается, что

поэтому

Поэтому имеем следующие разбиения множеств

и

на попарно непересекающиеся множества:

Для

будет биекцией из

в

, а

будет биекцией из

в

. Тогда,

также будут биекциями. Заметим также, что

(как и

) будет биекцией из

в

. Таким образом у нас получается великолепная наглядная картинка, все стрелочки в которой являются биекциями между соответствующими прямоугольниками:
--(3,6);
\draw [shift={(0,-2)}](0,6)--(3,6);
\draw [shift={(0,-3)}](0,6)--(3,6);
\draw [shift={(0,-4.5)}](0,6)--(3,6);
\node at (1.5,7.5) {$A$};
\draw (7,0) rectangle (10,7);
\draw (7,6)--(10,6);
\draw [shift={(0,-1)}](7,6)--(10,6);
\draw [shift={(0,-2)}](7,6)--(10,6);
\draw [shift={(0,-3)}](7,6)--(10,6);
\draw [shift={(0,-4.5)}](7,6)--(10,6);
\node at (8.5,7.5) {$B$};
\node at (1.5,6.5) {$A_0\setminus A_1$};
\node at (1.5,5.5) {$A_1\setminus A_2$};
\node at (1.5,4.5) {$A_2\setminus A_3$};
\node at (1.5,3.5) {$A_3\setminus A_4$};
\node at (1.5,2.25) {$\dots$};
\node at (1.5,0.75) {$\cap_{i=0}^{\infty}A_i$};
\node at (8.5,6.5) {$B_0\setminus B_1$};
\node at (8.5,5.5) {$B_1\setminus B_2$};
\node at (8.5,4.5) {$B_2\setminus B_3$};
\node at (8.5,3.5) {$B_3\setminus B_4$};
\node at (8.5,2.25) {$\dots$};
\node at (8.5,0.75) {$\cap_{i=0}^{\infty}B_i$};
\draw [->, thick, green] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
\draw [->, thick, green, shift={(0,-1)}] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
\draw [->, thick, green, shift={(0,-2)}] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
\draw [->, thick, green, shift={(0,-3)}] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
\draw [->, thick, green] (3.2,0.95)--(6.8,0.95);
\draw [->, thick, red, shift={(10,0)}, xscale=-1] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
\draw [->, thick, red, shift={(10,-1)}, xscale=-1] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
\draw [->, thick, red, shift={(10,-2)}, xscale=-1] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
\draw [->, thick, red, shift={(10,-3)}, xscale=-1] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
\draw [<-, thick, red] (3.2,0.55)--(6.8,0.55);
\node [green]at (4.5,6.7) {$f$};
\node [green]at (4.5,5.7) {$f$};
\node [green]at (4.5,4.7) {$f$};
\node [green]at (4.5,3.7) {$f$};
\node [green]at (4.9,1.3) {$f$};
\node [red]at (5.6,6.65) {$g$};
\node [red]at (5.6,5.65) {$g$};
\node [red]at (5.6,4.65) {$g$};
\node [red]at (5.6,3.65) {$g$};
\node [red]at (5.3,0.3) {$g$};
\end{tikzpicture} \begin{tikzpicture}
\draw (0,0) rectangle (3,7);
\draw (0,6)--(3,6);
\draw [shift={(0,-1)}](0,6)--(3,6);
\draw [shift={(0,-2)}](0,6)--(3,6);
\draw [shift={(0,-3)}](0,6)--(3,6);
\draw [shift={(0,-4.5)}](0,6)--(3,6);
\node at (1.5,7.5) {$A$};
\draw (7,0) rectangle (10,7);
\draw (7,6)--(10,6);
\draw [shift={(0,-1)}](7,6)--(10,6);
\draw [shift={(0,-2)}](7,6)--(10,6);
\draw [shift={(0,-3)}](7,6)--(10,6);
\draw [shift={(0,-4.5)}](7,6)--(10,6);
\node at (8.5,7.5) {$B$};
\node at (1.5,6.5) {$A_0\setminus A_1$};
\node at (1.5,5.5) {$A_1\setminus A_2$};
\node at (1.5,4.5) {$A_2\setminus A_3$};
\node at (1.5,3.5) {$A_3\setminus A_4$};
\node at (1.5,2.25) {$\dots$};
\node at (1.5,0.75) {$\cap_{i=0}^{\infty}A_i$};
\node at (8.5,6.5) {$B_0\setminus B_1$};
\node at (8.5,5.5) {$B_1\setminus B_2$};
\node at (8.5,4.5) {$B_2\setminus B_3$};
\node at (8.5,3.5) {$B_3\setminus B_4$};
\node at (8.5,2.25) {$\dots$};
\node at (8.5,0.75) {$\cap_{i=0}^{\infty}B_i$};
\draw [->, thick, green] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
\draw [->, thick, green, shift={(0,-1)}] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
\draw [->, thick, green, shift={(0,-2)}] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
\draw [->, thick, green, shift={(0,-3)}] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
\draw [->, thick, green] (3.2,0.95)--(6.8,0.95);
\draw [->, thick, red, shift={(10,0)}, xscale=-1] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
\draw [->, thick, red, shift={(10,-1)}, xscale=-1] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
\draw [->, thick, red, shift={(10,-2)}, xscale=-1] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
\draw [->, thick, red, shift={(10,-3)}, xscale=-1] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
\draw [<-, thick, red] (3.2,0.55)--(6.8,0.55);
\node [green]at (4.5,6.7) {$f$};
\node [green]at (4.5,5.7) {$f$};
\node [green]at (4.5,4.7) {$f$};
\node [green]at (4.5,3.7) {$f$};
\node [green]at (4.9,1.3) {$f$};
\node [red]at (5.6,6.65) {$g$};
\node [red]at (5.6,5.65) {$g$};
\node [red]at (5.6,4.65) {$g$};
\node [red]at (5.6,3.65) {$g$};
\node [red]at (5.3,0.3) {$g$};
\end{tikzpicture}](https://dxdy-02.korotkov.co.uk/f/1/5/4/1547df58302a1c33f4f7c5c0d6ef1c4f82.png)
С картинкой теперь все легко :) Последовательность множеств

, о которой шла речь - это

("прыгаем" по разбиению через одно множество) :

Теперь попарная непересекаемость множеств

вопросов не вызывает.
Также из картинки видно, как нужно переставить стрелочки, чтобы сконструировать биекцию из

в

:
--(3,6);
\draw [shift={(0,-2)}](0,6)--(3,6);
\draw [shift={(0,-3)}](0,6)--(3,6);
\draw [shift={(0,-4.5)}](0,6)--(3,6);
\node at (1.5,7.5) {$A$};
\draw (7,0) rectangle (10,7);
\draw (7,6)--(10,6);
\draw [shift={(0,-1)}](7,6)--(10,6);
\draw [shift={(0,-2)}](7,6)--(10,6);
\draw [shift={(0,-3)}](7,6)--(10,6);
\draw [shift={(0,-4.5)}](7,6)--(10,6);
\node at (8.5,7.5) {$B$};
\node at (1.5,6.5) {$A_0\setminus A_1$};
\node at (1.5,5.5) {$A_1\setminus A_2$};
\node at (1.5,4.5) {$A_2\setminus A_3$};
\node at (1.5,3.5) {$A_3\setminus A_4$};
\node at (1.5,2.25) {$\dots$};
\node at (1.5,0.75) {$\cap_{i=0}^{\infty}A_i$};
\node at (8.5,6.5) {$B_0\setminus B_1$};
\node at (8.5,5.5) {$B_1\setminus B_2$};
\node at (8.5,4.5) {$B_2\setminus B_3$};
\node at (8.5,3.5) {$B_3\setminus B_4$};
\node at (8.5,2.25) {$\dots$};
\node at (8.5,0.75) {$\cap_{i=0}^{\infty}B_i$};
\draw [->, thick, green] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
%\draw [->, thick, green, shift={(0,-1)}] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
\draw [->, thick, green, shift={(0,-2)}] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
%\draw [->, thick, green, shift={(0,-3)}] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
\draw [->, thick, green] (3.2,0.95)--(6.8,0.95);
\draw [<-, thick, red, shift={(10,0)}, xscale=-1] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
%\draw [->, thick, red, shift={(10,-1)}, xscale=-1] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
\draw [<-, thick, red, shift={(10,-2)}, xscale=-1] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
%\draw [->, thick, red, shift={(10,-3)}, xscale=-1] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
%\draw [<-, thick, red] (3.2,0.55)--(6.8,0.55);
\node [green]at (4.5,6.7) {$f$};
\node [green]at (4.5,5.7) {$f$};
\node [green]at (4.5,4.7) {$f$};
\node [green]at (4.5,3.7) {$f$};
\node [green]at (4.9,1.3) {$f$};
\node [red]at (5.6,6.65) {$g^{-1}$};
%\node [red]at (5.6,5.65) {$g$};
\node [red]at (5.6,4.65) {$g^{-1}$};
%\node [red]at (5.6,3.65) {$g^$};
%\node [red]at (5.3,0.3) {$g$};
\node at (5,2.25) {$\dots$};
\end{tikzpicture} \begin{tikzpicture}
\draw (0,0) rectangle (3,7);
\draw (0,6)--(3,6);
\draw [shift={(0,-1)}](0,6)--(3,6);
\draw [shift={(0,-2)}](0,6)--(3,6);
\draw [shift={(0,-3)}](0,6)--(3,6);
\draw [shift={(0,-4.5)}](0,6)--(3,6);
\node at (1.5,7.5) {$A$};
\draw (7,0) rectangle (10,7);
\draw (7,6)--(10,6);
\draw [shift={(0,-1)}](7,6)--(10,6);
\draw [shift={(0,-2)}](7,6)--(10,6);
\draw [shift={(0,-3)}](7,6)--(10,6);
\draw [shift={(0,-4.5)}](7,6)--(10,6);
\node at (8.5,7.5) {$B$};
\node at (1.5,6.5) {$A_0\setminus A_1$};
\node at (1.5,5.5) {$A_1\setminus A_2$};
\node at (1.5,4.5) {$A_2\setminus A_3$};
\node at (1.5,3.5) {$A_3\setminus A_4$};
\node at (1.5,2.25) {$\dots$};
\node at (1.5,0.75) {$\cap_{i=0}^{\infty}A_i$};
\node at (8.5,6.5) {$B_0\setminus B_1$};
\node at (8.5,5.5) {$B_1\setminus B_2$};
\node at (8.5,4.5) {$B_2\setminus B_3$};
\node at (8.5,3.5) {$B_3\setminus B_4$};
\node at (8.5,2.25) {$\dots$};
\node at (8.5,0.75) {$\cap_{i=0}^{\infty}B_i$};
\draw [->, thick, green] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
%\draw [->, thick, green, shift={(0,-1)}] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
\draw [->, thick, green, shift={(0,-2)}] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
%\draw [->, thick, green, shift={(0,-3)}] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
\draw [->, thick, green] (3.2,0.95)--(6.8,0.95);
\draw [<-, thick, red, shift={(10,0)}, xscale=-1] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
%\draw [->, thick, red, shift={(10,-1)}, xscale=-1] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
\draw [<-, thick, red, shift={(10,-2)}, xscale=-1] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
%\draw [->, thick, red, shift={(10,-3)}, xscale=-1] (3.2,6.5).. controls (5,6.4) and (5.6,6.2).. (6.8,5.5);
%\draw [<-, thick, red] (3.2,0.55)--(6.8,0.55);
\node [green]at (4.5,6.7) {$f$};
\node [green]at (4.5,5.7) {$f$};
\node [green]at (4.5,4.7) {$f$};
\node [green]at (4.5,3.7) {$f$};
\node [green]at (4.9,1.3) {$f$};
\node [red]at (5.6,6.65) {$g^{-1}$};
%\node [red]at (5.6,5.65) {$g$};
\node [red]at (5.6,4.65) {$g^{-1}$};
%\node [red]at (5.6,3.65) {$g^$};
%\node [red]at (5.3,0.3) {$g$};
\node at (5,2.25) {$\dots$};
\end{tikzpicture}](https://dxdy-03.korotkov.co.uk/f/a/d/6/ad6e0b58f13c5a28a5750ab3b381368182.png)