Weyl's lemma tells us that if we repeatedly advance a dial an irrational portion of a full circle, it will not "prefer" any portion of the circle. Obvious, suspected only in the Middle Ages (recall the Greeks knew about irrationals!), but only proved in the 20th century. A proof of Weyl's lemma requires Fourier analysis or some equivalent.

**Lemma.** Let α be an irrational number, and define the sequence x_{n}={n*α}=n*α-[n*α] of the fractional parts of multiples of α. Then x_{n} is uniformly distributed.

This is intuitively obvious, and considerably stronger than Kronecker's lemma.