Final answer:
Cantor's theorem can be used to prove that there is no infinite set with smaller cardinality than the set of natural numbers and the set of real numbers.
Step-by-step explanation:
In set theory, the size (or cardinality) of a set is defined by the concept of cardinal numbers. In this case, the question is asking to prove that there is no infinite set A such that the cardinality of A is less than the cardinality of the set of natural numbers (N) and also less than the cardinality of the set of real numbers (aleph-null or א0).
To prove this, we can use Cantor's theorem, which states that for any set A, the cardinality of the power set of A is strictly greater than the cardinality of A.
Now, assume that there exists an infinite set A with |A| < |N| = א0. The power set of A, denoted as P(A), is the set of all subsets of A. Since A is infinite, P(A) is also infinite. However, according to Cantor's theorem, the cardinality of P(A) is strictly greater than the cardinality of A. This contradicts our assumption, therefore, proving that the statement is false.