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First, don’t be concerned about the statements on cardinality.
Once you have the basic idea down, all the rest is easy.
Secondly, the above is not the Schroeder-Bernstein. But nonetheless it is easy to prove anything about finite sets.
The basic idea of a finite set is: Any finite set can be listed using positive integers as subscripts.
Thus if $\displaystyle n = \left| X \right| \le \left| Y \right| = m$ then we can write $\displaystyle X = \left\{ {x_1 ,x_2 ,x_3 , \cdots ,x_n } \right\}\,\& \,Y = \left\{ {y_1 ,y_2 ,y_3 , \cdots y_m } \right\}$.
If we define $\displaystyle f:X \mapsto Y$ by $\displaystyle f\left( {x_j } \right) = y_j $ the result follows easily.