Hi Isabel,

You are right. I misunderstood the proof presented on the video with the «old-fashion» Euclides proof that seems to be somehow slightly diferent (I didn't watch closely the video and thought there was only one version of the proof).

I found this page where anybody can check both versions of the same demonstration

Euclid's Proof of the Infinitude of Primes (c. 300 BC)

Thanks for your post and your comment!