Evaluating $|E_K(x)| = s(x) - \sum_{k=1}^K \dfrac{\sin 2 \pi k x}{\pi k}$ on $0<x \le 1/2$

Question

The following is from "Multiplicative number theory I: Classical theory" by Hugh L. Montgomery, Robert C. Vaughan:

I could understand the full proof but the last paragraph, i.e. to show that $|E_K(x)| \le 1/2.$

I might not receive any answer if I ask for re-write the last paragraph so I ask my questions in details:

Question 1: For $1/(2K + 1) ≤ x ≤ 1/2$ we have $\sin πx ≥ 2x$. But how does this imply $|E_K(x)| \le 1/2$?

Question 2: For $0 ≤ u ≤ π$, we have $0 < \sin u < u$. But how does this implies $x − 1/2 ≤ E_K (x) ≤ (2K + 1)x − 1/2$ for $0 ≤ x ≤ 1/(2K + 1)$? I made calculations several times but I failed to achieve the result.

Question 3: How does $x − 1/2 ≤ E_K (x) ≤ (2K + 1)x − 1/2$ for $0 ≤ x ≤ 1/(2K + 1)$ imply $|E_K(x)| \le 1/2$?