Call a vector X = (x, y) in R2 "resonant" if it satisfies an equation
of the form ax + by + c = 0 where a, b, c are integers, not all zero.
Otherwise X is non-resonant. Use Baire theory to show that the set
of "non-resonant" vectors is dense in R2.
All I have so far is that a vector is non-resonant if it satisfies x not rational, y not rational or x /= q1 + q2*y, where q1 and q1 are rational.
Note: Apply Baire's theory is a nice way of saying that we have to show the set is thick; that is, the intersection of countably many open and dense sets.
For any , not all 0,you can easily check that is open. If you show that is dense, you conclude because the set of all non resonant vectors is the countable intersection of all the . To check this density you have to assume and show that there is "as near as desired" from such that .