Question on Dirichlet Character conductor (please read description)

Question

So I'm reading a paper on "The Dirichlet L-Series" (link below) and came across this:

We'll define the Dirichlet Character:

$$\chi \space : \space (\mathbb{Z}/f\mathbb{Z})^\times \longrightarrow \mathbb{C}^\times$$

Such that $\space f \space$ is the conductor of $\space \chi$.

There is a statement made in the paper saying:

If $\space f \space$ is the conductor of $\space \chi \space$, then $\space \big|\sum_{i=1}^{n} \chi(i) \big| \le f \space$ for all $\space n \space$.

Can someone please explain why this is true?

I've been looking everywhere online and can't seem to find a proof for this.

Here is the link to the paper, the statement is in Page 2 at the start of the proof for Lemma 1.5:

https://www.math.ucdavis.edu/~osserman/classes/254a/lectures/22.pdf

Answer 1

As long as $\chi$ is non-trivial $\sum_{i=1}^f\chi(i)=0$. As $\chi$ has period $f$ then for $n=qf+r$ ($0\le r<f$) then $$\sum_{i=1}^n\chi(f)=\sum_{i=qf+1}^{qf+r}\chi(i).$$ This last sum has $<f$ terms.