Given positive real numbers $x$ and $\epsilon$, do there necessarily exist positive integers $p$ and $q$ such that $\left| qx - p \right| < \epsilon$?
Or geometrically: in a two-dimensional Cartesian plane, does every line through the origin come arbitrarily close to some other lattice point? (I realize that that's not exactly equivalent to the algebraic version, but if either is true than both are.)
I feel pretty certain that the answer must be "yes", but I'm not managing to prove it. My initial reaction was that it should be a direct consequence of the fact that the rationals are a dense subset of the reals; but that really only shows that we can satisfy the weaker condition that $\left| x - \frac{p}{q} \right| < \epsilon$, or equivalently, that $\left| qx - p \right| < q\epsilon$.
(This question was inspired by this recent answer on the Space Exploration Stack Exchange.)
Yes, this follows from a pigeonhole argument. https://en.wikipedia.org/wiki/Dirichlet%27s_approximation_theorem