Consider a injective function f: A --> A

A being a finite Set.

Lets assume that the function f is not Surjective:

Therefore the equality f(x)≠y does not holds for some 'y' .

But since each element of A must be mapped to an element of A, if there were an element y such that there doesn’t exist an element x in A such that f(x) = y, then one element of A must be mapped to two elements for the function f to exist.

Elements of A must be mapped to elements of A.But if f(x)≠y ,then for a function to be valid one element maps to 2 elements which is not possible in case of injective function.

Therefore if f:A--> is Injective then it is also Surjective.

Hence Proved!