In many textbooks the proof of Cauchy-Schwarz inequality needs to introduce a parameter $\lambda$ to take $\langle x+\lambda y,x+\lambda y \rangle$ as the first step. I am trying to proof the Cauchy-Schwarz inequality starting from the triangle inequality as follows:
By triangle inequality of norm: $\|x+y\| \le \|x\|+\|y\|$
Taking square on both sides: $\|x+y\|^2 \le (\|x\|+\|y\|)^2$
The norm induced by inner product: $\langle x+y,x+y \rangle \le (\|x\|+\|y\|)^2$
Expanding both sides: $\langle x,x \rangle + \langle x,y \rangle + \langle y,x \rangle + \langle y,y \rangle \le \|x\|^2 + 2\|x\|\|y\| + \|y\|^2$
Eliminating the square norms: $\langle x,y \rangle + \langle y,x \rangle \le 2\|x\|\|y\|$
By the property of inner product: $\langle x,y \rangle + \overline{\langle x,y \rangle} \le 2\|x\|\|y\|$
I already have the Cauchy-Schwarz inequality for $\mathbb{R}$-vector space, as $\langle x,y \rangle \equiv \overline{\langle x,y \rangle}$. But for $\mathbb{C}$-vector space, can we finally obtain the Cauchy-Schwarz inequality along this approach?