$ 1_x $
Assume that there are only a finite number of prime number p_1,p_2,p_3,......p_n. Then by using the fact from exercise 18 (Let p_1,p_2,p_3,....,p_n be primes. Then p_1p_2.....p_n +1 is divisible by none of these primes), p_1p_2p_3....p_n +1 is not divisible by any prime.) This means p_1p_2...p_n +1 (which is larger than our initial conditions) is itself prime. This contradicts the assumption that p_1,p_2,...p_n is the list of all primes.
~Angela
