I'm trying really hard to get good at making proofs. It is extremely hard for me, much more so than anything else I have ever done. I've tried to prove the title above but I don't know if my proof is logical.

lemma: "If n > 1 then there is a prime p such that p | n ."

n and k are integers.

proof:

Assume that there is no such p for n>1 such that

This would imply that

for any prime value of p.

This also implies that there are no primes that can be expressed as the division of two integers.

However if this was true, prime numbers would be irrational which they are not.

Therefore all numbers>1 are divisable by a prime?

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I'm not sure if this is logical. If it is not please don't post the proof, just give me a hint?