Telescoping increment lemma
= Telescoping increment lemma
If a continuous $g:[0,na]\to\mathbb R$ satisfies $g(na)-g(0)=na$, then the increments
$$
h(x)=g(x+a)-g(x)-a
$$
satisfy $\sum_{k=0}^{n-1}h(ka)=0$. Continuity forces $h$ to vanish somewhere on $[0,(n-1)a]$.