Let $\alpha$ be an irrational number and $\beta$ be an arbitrary real number, Prove that there are infinitely many pair of integers $(x,y)$ with $x\in\mathbb{N}$ such that:
$$|x\alpha-y-\beta|<\frac3x$$
This theorem is due to polish mathematician Leopold Kronecker.
There is even sharper results by making use of geometry of numbers.
Let $N$ be an arbitrary natural number. By Dirichlet's approximation theorem there exist two relatively prime integers $p,q$ with $q\gt 2N$ such that, $$|q\alpha-p|\lt\frac1q$$ Let $Q$ be the integer or one of the integers such that, $$|Q-q\beta|\le\frac12$$ By using of Bézout's identity there exist two integers $x_0,y_0$ such that, $$px_0-qy_0=Q$$ Because $p,q$ were relatively prime integers. We may suppose that, $$|x_0|\le\frac12q$$ Because the set $\{px_0:|x_0|\le\frac12q\}$ contains all integers modulo $q$. By now, we have, $$|q(x_0\alpha-y_0-\beta)|=|x_0(q\alpha-p)+px_0-qy_0-q\beta|=|x_0(q\alpha-p)+Q-q\beta|\le |x_0(q\alpha-p)|+|Q-q\beta|\lt \frac12q\cdot \frac1q+\frac12=1$$ Taking $x=x_0+q$ and $y=y_0+p$ we have $N\lt\frac12q\le|x|\le\frac32q$. So, $$|x(\alpha-y-\beta)|\le|x_0\alpha-y_0-\beta|+|q\alpha-p|\lt\frac1q+\frac1q=\frac2q\le\frac3{|x|}$$ This proves the theorem!