I was looking through a proof and i stumbled upon this inequality :
$\displaystyle | e^{ia}-1 | \leq |a |$
$\displaystyle a \in \mathbb{R}$
but i cannot convince myself why it is true.
Can you think about any hint?
thnx
For a second suppose that $\displaystyle a\geqslant 0$ then $\displaystyle \left|e^{ia}-1\right|=\sqrt{\left(\cos(a)-1\right)^2+\sin^2(a)}=$$\displaystyle \sqrt{\cos^2(a)+\sin^2(a)+1-2\cos(a)}=\sqrt{2}\sqrt{1-\cos(a)}=2\sin\left(\frac{a}{2}\right)\leqslant 2\frac{a}{2}=a$