For each positive integer $n$ , show that there are more than $n$ positive primes. Suppose a picked the positive integer 7. Of course there are more than 7 prime numbers in Z. I just don't know how to show that. How do you show that?
 A: The Wikipedia article on "Euclid's Theorem" (https://en.wikipedia.org/wiki/Euclid%27s_theorem) gives several proofs of the theorem which are very accessible. Some of them were entirely new to me. (Does everyone but me know of the beautiful proof by Furstenberg?)
P.S. I can't comment on this site, but i just saw your question about the contradiction at the heart of Euclid's proof. It's very simple: at the beginning you assume that there are k primes -- what is meant is that there are EXACTLY k primes (k is some particular number). But then Euclid produces a number which has a non-zero remainder when divided by every one of the k primes. If there were only exactly k primes, then that new number would have to be divisible by one of them. But it's not -- so either there is some OTHER number, not among the list of k primes, that is a prime divider, or that number itself is prime. Either way, it's just not true that there are exactly k primes. 
P.P.S. Not that I particularly care, but I notice that within seconds of posting this answer, there were two down votes. I found that strange: I actually answered his actual question, which was "how do you show that?" There are many many ways to show the result, only one of which is ascribed to Euclid. I gave a link to a wide variety of ways to "show" that. And then I answered his particular question about how Euclid showed that. I can only assume that the down votes were because the original hyperlink led to some error page. Well, that error page was erroneous (you can't believe everything you read in Wikipedia :)) -- there IS an article in Wikipedia with the exact name I gave. In any case, the link is now correct. Perhaps the hater-voters can unhate their previous votes, or explain the hate. Thx!
