Let $\alpha$ and $\beta$ be two irrational numbers. It is an interesting exercise to prove using the Pigeon Hole Principle that the sequences $\{n \alpha \}_{n = 1}^{\infty}$ and $\{ n \beta \}_{n = 1}^{\infty}$ are both dense in the unit interval modulo $1$. Can we use this fact (or something else) to prove the following?
Let $\epsilon > 0$. There exists a positive integer $M$ such that the following inequality holds true: $$0< \{ M \alpha \}, \{M \beta\} < \epsilon $$ where $\{ \cdot \}$ denotes the fractional part.
Theorem of Kronecker. Thm. 3.4 in Diophantine Approximations by Ivan Niven.
If $\alpha, \beta $ are irrational numbers such that $1,\alpha, \beta $ are linearly independent over the field $\mathbb Q.$ Then, for $m=1,2,3 \ldots,$ $$ \left( \{ m \alpha\}, \{ m \beta \} \right) $$ make a dense set in the unit square
In the notes at end of chapter, he says these points are uniformly distributed in the square. I see, any dimension, he refers to Cassels (1957), chapter 4; title An Introduction to Diophantine Approximation.