I will quote Wikipedia while doing this.
This is the only proof I am familar with (I know there are many).
You need to be familar with the Zeta function,
It diverges for (integral p-series test).
Converges for .
You also need to be familar with one of the most elegant formulas, "Euler-Product formula".
It states we can factorize the zeta function as,
The final concept you need to know is how to work with infinite products. If you an infinite product of positive terms,
You can take the natural logarithm to obtain,
And then work with convergence with the infinite series!
Now we can begin with Euler's proof.
By Euler's product formula.
Expressing the product as a sum we have, (natural logarithms are positive)
Exponent rule for logarithms,
Infinite series for logarithm ( )
But this is (absolute convergenct thus we can rearrange),
This is strictly less than,
Geomteric series ( ),
Then, by the dominance rule,
Is bounded by this sum.
Which is not possible because this is the infamous harmonic series which diverges.
Note, since the harmonic series behaves like and the sum of reciprical of primes is the natural logarithm of that we can say,
(Note not a prove, an observation).
Thus, it diverges really really slowly.
Thus the number of primes is infinite.
Because otherwise the reciprocal sum would converge.