Answer
Assume there is a finite subset of S called A, if S is infinite, then every element of the proper subset A can be assigned a unique element in S.
If you add more elements from S to A then since S is infinite, you can just shift over the set by as much as you are adding to the set to give each element in A a unique image in S, this can be done an infinite amount of times which proves that S is an infinite set.
Note that we proved that S is one to one no matter how many elements from S are added to A.
Work Step by Step
Assume there is a finite subset of $S$ called $A$, if $S$ is infinite, then every element of the proper subset $A$ can be assigned a unique element in $S$.
If you add more elements from $S$ to $A$ then since $S$ is infinite, you can just shift over the set by as much as you are adding to the set to give each element in $A$ a unique image in $S$, this can be done an infinite amount of times which proves that $S$ is an infinite set.
Note that we proved that $S$ is one to one no matter how many elements from $S$ are added to $A$.