Vinogradov's Theorem/Minor Arcs/Lemma 1

Lemma
For $\beta \in \R$, define $\|\beta\| = \min\{|n - \beta| : n \in \Z\}$.

Then for any $\alpha \in \R$,


 * $\displaystyle \left| \sum_{k = N_1}^{N_2} e(\alpha k) \right| \leq \min\left\{ N_2 - N_1, \frac1{2\|\alpha\|} \right\}$