Distance to the nearest integer $<r$ implication

Question

The following is an adaptation of a result in the middle of a proof I am reading. The result is not obvious to me and I don’t know how they were able to conclude it.

I define $\langle a \rangle$ to be the distance from $a$ to the nearest integer for $a\in \mathbb{R}$.

Let $\alpha$ be irrational; $q$ be a natural number and $r \in [0,\frac{1}{2}]$. Then:

$$\langle q\alpha \rangle < r \implies \mid \alpha - \frac{p}{q} \mid < \frac{r}{q}$$ for some $p\in \mathbb{N}$

Answer 1

Let the integer closest to $ q \alpha $ be $p$. Then the result follows.

$$ \langle q\alpha \rangle < r \Rightarrow |q\alpha - p | < r \Rightarrow |\alpha - \frac{p}{q} | < \frac{ r}{q}.$$