In this problem, we will explore how the cardinality of a subset S X relates to the cardinality of a finite set X.

(i) Explain why |S| |X| for every subset S X when |X| = 1.

(ii) Assume we know that if S , then |S| n. Explain why we can show that if T , then |T| n + 1.

iii) Explain why parts (i) and (ii) imply that for every n N, every subset of n is finite and has cardinality less than n + 1