Prove that every set that has an infinite subset is infinite.
Also, prove that every subset of a finite set is finite.
Suppose that $\displaystyle B$ is an infinite and $\displaystyle B \subseteq A$ we want to show that $\displaystyle A$ is infinite.
By the definition $\displaystyle \left( {\exists C} \right)\left[ {C \varsubsetneq B} \right]$ and there is a bijection such that $\displaystyle f:B \mapsto C$.
It should be clear that $\displaystyle D = \left( {A\backslash B} \right) \cup C$ is a proper subset of $\displaystyle A$.
Define a function $\displaystyle g:A \mapsto D$ as $\displaystyle g(x) = \left\{ {\begin{array}{rl} {f(x),} & {x \in B} \\ {x,} & {x \in A\backslash B} \\ \end{array} } \right. $
Now your task is to show that $\displaystyle g$ is a bijection.