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
