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Assume there are finitely many prime number,

P1, P2, P3,.....,Pn.

Let q=(P1*P2*P3*.....*Pn)+1

Then, for all 1≤i≤n, Pi does not divide q, so therefore there must be at least Pn more prime, Pn+1.

Here we reach a contradiction since we assumed there were only n primes. Therefore our assumption is in error and there are infinitely many primes. #

I don't understand the portion in red. Can anyone please explain it in simple terms?

Thanks.