How to prove triangle inequality for euclidean norm on complex number?

Question

We were asked to show that when: $\displaystyle \Vert Z\Vert = \left(\sum_{k=1}^{n} (x_k+iy_k)(x_k-iy_k)\right)^{1/2}$ that $\Vert Z+W\Vert \leq \Vert Z\Vert+\Vert W\Vert$ whenever $Z$ and $W$ are vectors in complex numbers holds.

Can someone help me to prove this please?

Answer 1

Hint: the Cauchy-Schwarz inequality.

Answer 2

Assuming Cauchy-Schwarz's inequality $|\langle z,w\rangle| \leq \|z\|\|w\|$, you have: $$\begin{align} \|z+w\|^2 &= (z+w)\overline{(z+w)} \\ &= (z+w)(\overline{z}+\overline{w}) \\ &= z\overline{z}+z\overline{w}+\overline{z}w+w\overline{w} \\ &= \|z\|^2 + 2\,{\rm Re}(z\overline{w}) + \|w\|^2 \\ &\leq \|z\|^2+2|\langle z,w\rangle|+\|w\|^2 \\ &\leq \|z\|^2 + 2\|z\|\|w\|+\|w\|^2 \\ &= (\|z\|+\|w\|)^2, \end{align}$$ and take roots.