I am having difficulty in understanding the proof of Behmann of Infinitude of primes. Can someone please explain the last part 'The proof is concluded by noticing....' which is in page $178$?
Any help would be appreciated. Thanks in advance.
2026-04-05 10:05:33.1775383533
Behmann's proof of Infinitude of primes.
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Suppose there are only $m$ primes. The expression
$$\frac{p_i}{p_i-1} = \frac{1}{1-\frac{1}{p_i}} = 1 + \frac{1}{p_i} + \frac{1}{p_i^2} +\cdots,$$
by geometric series.
So if the right hand side is multiplied out, you get every possible unit fraction. So the right hand side must be equal to the harmonic series. But we've just shown that the harmonic series is strictly larger. So there must be more than $m$ primes.