Famous Theorems of Mathematics/Sum of the reciprocals of the primes diverges

The series diverges, where is the nth prime number. This proof is a proof by contradiction. Proof Assume for the sake of contradiction that converges. We pick a large enough number to make a number such that We create a new geometric series This geometric series converges because Expanding this series yields When expanding each term, this series will contain all fractions of where only consists of prime factors in the set although there will be many duplicates of this fraction. Therefore, and thus this series converges. Consider a new series The denominator of each term in this series, , must only have prime factors in the set This is because if it did have a prime factor less than , say , then is an integer and thus is an integer, but this is impossible. So, must only have prime factors in the set which means and thus this new series converges. However, This would mean that also converges, but this is just the harmonic series, which diverges. A contradiction. ∎