Let $$f(x)=\sum_n^\infty c^n\langle d^nx\rangle $$ where $\langle x \rangle=\min _{n\in\mathbb{Z}}|x-2n|$, i.e. the distance to the closest even integer, $c,d>0$. Prove that if $c\cdot d<1$ then $f$ is Lipschitz on $\mathbb{R}$.
I know that in order to prove a function is Lipschitz I need to find a constant (or at least prove that one exists) satisfying $\forall x,y$ $$ |f(x)-f(y)|<M|x-y| $$ I also know that $\langle x \rangle \leq 1$ so I can pretty much get rid of the $x$ values whenever I want, but I honestly have no idea how to even start proving that function is Lipschitz. Any hints?
Hint: start my showing $|\langle t - s \rangle| \le |t - s|$.