Prove that there are infinitely many primes of the form $3n - 1$.
For the purpose of contradiction assume there are a finite number $k$ of primes $p_1, p_2, \ldots, p_k$ of the form $3n-1$.
Consider the number
$$ N=3\times p_1 \times p_2 \times \ldots p_k - 1 $$
which is of the form $3n-1$.
Since $N>1$ it has prime factors. Any such prime factor, let's call it $q$, is not 3 or any of the $p_1, p_2, \ldots, p_k$ because that would mean $p_i \mid 1$ which is not possible.
Is $q$ of the form $3n-1$? Since $N \equiv -1 \pmod 3$ that means an odd number of the prime factors of $N$ are congruent to $-1 \pmod 3$, that is, of the form $3n-1$. That is, at least one.
The existence of at least one prime factor of $N$ of the form $3n-1$ which is not one of the $p_1, p_2, \ldots, p_k$ contradicts the assumption there are a finite number of primes of the form $3n-1$.