Let's fill in the blanks for the proof in Problem 9 based on the information provided and the typical structure of this proof.

Given Information:

1. Theorem 1 states that if $b$ is a natural number, then $\gcd(b, b+1) = 1$, meaning $b$ and $b+1$ are coprime.
2. Theorem 2 provides a criterion for an infinite set: if for each finite subset $S$ of a set $B$, there exists an element $x \in B$ such that $x \notin S$, then $B$ is infinite.
3. Theorem 3 is the statement we need to prove, that there are infinitely many prime numbers.

Proof Outline with Filled Blanks:

Let's proceed step-by-step and fill each blank in the proof.

1. Let $\mathbb{P}$ be the set of prime numbers.

   • This is the set of all primes, which we are trying to show is infinite.

2. Let $Q$ be a finite subset of the set $\mathbb{P}$. Denote the elements of $Q$ by $p_1, p_2, \dots, p_n$ and let $q = p_1 \cdot p_2 \cdots p_n$.

   • Here, we consider a finite subset $Q$ of prime numbers and define $q$ as the product of all primes in $Q$.

3. By Theorem 1, $q$ and $q + 1$ are coprime.

   • According to Theorem 1, $q$ (product of primes in $Q$) and $q+1$ are coprime.

4. So there is at least one prime number that divides $q + 1$ but not $q$.

   • There must be a prime factor of $q+1$ that is not in the original set $Q$, because $q+1$ is not divisible by any of the primes in $Q$.

5. Let's call this prime number $t$. Because $t$ does not divide $q$, we have that $t$ is not an element of $Q$.

   • This new prime $t$ divides $q+1$, proving that there is always a prime not in $Q$.

6. So we have shown that for any finite set of prime numbers $Q$, we can find another prime number that is not in the set $Q$. Thus, by Theorem 2, we have that $\mathbb{P}$ is infinite.

   • By Theorem 2, since we can always find a new prime outside any finite subset, the set of primes $\mathbb{P}$ must be infinite.

Final Answer for Each Blank:

1. $\mathbb{P}$ = prime numbers.
2. $Q$ = finite subset.
3. coprime.
4. divides.
5. not.
6. infinite.

Would you like additional details on any part of this proof or have any questions?

Here are some related questions to expand your understanding:

1. Why is $q+1$ guaranteed to have a prime factor not in $Q$?
2. How does Theorem 2 guarantee the infiniteness of $\mathbb{P}$?
3. Can you explain why $q$ and $q+1$ are coprime for any finite set $Q$?
4. What would happen if there were only finitely many primes?
5. How does this proof relate to Euclid’s original proof of the infiniteness of primes?

Tip: When proving a set is infinite, demonstrating that new elements can always be found outside any finite subset is often a strong approach.