I suggest this mapping: .
Of course is finite.
It is easy to show that is one to one.
Here are theorems that you need to do these sorts of cardinality problems.
T1 There is an injection from A to B if and only if there is a surjection from B to A.
T2 If there is an injection from A to B then the cardinality of A at most the cardinality of B.
T3 If there is an surjection from B to A then the cardinality of B at least the cardinality of A.