An assignment I am working on (Problem 4-4 in Lee's Introduction to Smooth Manifolds) reduces to the following problem. Given $\alpha, \beta, \varepsilon\in\mathbb{R}$, with $\alpha$ irrational and $\varepsilon > 0$, show there exist two integers, $m, n$ such that $$|m - \alpha \cdot n + \beta| < \epsilon.$$
If $\beta\in \mathbb{Z}$, then we could simply use Dirichlet's approximation theorem, but I've not been able to prove it in general.
One thing I've tried is approximating $\beta$ as a rational number $\tilde \beta = p/q \in \mathbb{Q}$, which leads to $|m - \alpha n + p/q | < \epsilon$ which can be manipulated into $|qm + p - \alpha q n | < \epsilon$. We can then let $\tilde m = qm + p$ and $\tilde n = q n$, and apply DAT to show $\tilde m$ and $\tilde n$ exist such that $|\tilde m + \alpha \tilde n| < q\epsilon$. One problem with this approach, though, is that when we solve for $m = \frac{\tilde m - p}{q}$ we aren't guaranteed to get an integer.
As suggested by Jyrki Lahtonen in the comments above, we can use Kronecker's Approximation Theorem, which, according to Wolfram MathWorld, is as follows for the one-dimensional case.
For any $\alpha, \beta, \epsilon \in \mathbb{R}$, with $\alpha$ irrational and $\epsilon > 0$, then there exists integers $m$ and $n$ with $n>0$, such that $$|m - \alpha n + \beta| < \epsilon.$$