Question 515378
We'll prove (b) first.  If T is an infinite set, and T is a subset of S, then S contains an infinite set, and therefore must be an infinite set itself.  There are more sophisticated ways to show this, but this is a perfectly reasonable proof.

Now notice that (a) is the contrapositive of (b).  Recall that the contrapositive of a statement "if p is true, then q is true" is "if q is not true, then p is not true."  Another way to state (a) is "if S is not an infinite set, then T is not an infinite set."  So (a) is the contrapositive of (b).  Now, we know that if an implication like (b) is true, then its contrapositive is also true, and vice versa.  So by proving (b) and showing that (a) is its contrapositive, we have proven (a).