Let f be a one-to-one function from A to B, with B being finite. Prove that A is finite.
Suppose that $\displaystyle f:A \mapsto B$ is injective.
Note that means $\displaystyle x \in A,\,y \in A\;\& \,x \ne y\, \Rightarrow \,f(x) \ne f(y)$
Using that property; $\displaystyle \left( {\forall a \in A} \right)\left\{ {\{ f(a)\} } \right\}$ is collection of pair-wise disjoint subsets of $\displaystyle B$.
If $\displaystyle B$ is finite then what can you say about $\displaystyle A$?