Arithmetic progression

An arithmetic progression is a sequence $a,a+r,\dots ,a+(m-1)r$ with constant successive difference $r$. It is nonconstant when $r\ne 0$.

Colour each interval $[2^j,2^{j+1})\cap \mathbb{N}$ by the parity of $j$. Every infinite arithmetic progression of positive common difference d meets every sufficiently large such interval, since the interval has length greater than d. Both colours occur infinitely often. Thus neither colour class contains an infinite arithmetic progression, despite the finite-length conclusion of the Van der Waerden theorem.

If all residues of a modular arithmetic progression have integer representatives in an interval of diameter less than half the modulus, the representatives form an integer arithmetic progression. Consecutive differences lie in an interval shorter than the modulus, so the common modular difference has a unique possible integer representative.

The common difference of an arithmetic progression is the fixed difference between consecutive terms.