The Cone Lemma. If a system of homogenous linear equations with integer coefficients has a positive real solution, then it also has a positive integer solution.
This is proved in Proofs from THE BOOK, 6th edition, p.70. I don't find the proof easy to understand, and I wonder if anyone can offer some intuition on why the lemma holds.
For example, does it still hold if the two instances of "positive" are removed? What role does the homogenous assumption play?