Proving the Infinitude of Primes: Euler's Proof and Its Limitations

[MostlyConfusd]

Infinite primes proof?

Someone told me Euler proved that there are infinitely many prime numbers by proving that the sum of their reciprocals is infinite.

I have one concern. How can you prove the infinitude of primes by this method without assuming the set to be infinite in the first place.

 

[sciencecrazy]

Search for zeta function by euler, over internet and how it's proved:

for s=1, LHS diverges.
So RHS must diverge as well, for s=1, which helps to deduce that rhs has infinite terms,ie. infinite number of primes.

So,you actually don't need to assume infinite number of primes before hand, but rather just that there are infinite number of positive integers.

 

[MostlyConfusd]

So, because the series diverges we can say there are infinitely many primes, but is that because the primes exhibit some uniformity in their distribution? my calc teacher has been over divergence and convergence several times and all that divergence seems to mean is that the denominator grows less quickly than that of a convergent function. Also, when a function converges, it doesn't prove the series is finite, so how can this be a helpful test?