It is not possible to prove that irrational numbers are infinite using the fact that prime numbers are infinite, because irrational numbers and prime numbers are not directly related.
Irrational numbers are numbers that cannot be expressed as a simple fraction (a ratio of two integers). Examples of irrational numbers include √2, π, and e. These numbers have an infinite number of decimal places and cannot be represented exactly as a finite decimal or fraction.
On the other hand, prime numbers are positive integers that have no positive integer divisors other than 1 and themselves. Prime numbers are infinite, meaning there is no largest prime number.
While it is true that both irrational numbers and prime numbers are infinite, there is no direct relationship between them. It is not possible to prove that irrational numbers are infinite using the fact that prime numbers are infinite, because the two concepts are distinct and unrelated.