The triangle inequality states that for any complex numbers $z_1,z_2,\ldots,z_n$ we have $$\lvert z_1+z_2+\cdots+z_n\rvert\leqslant\lvert z_1\rvert+\lvert z_2\rvert+\cdots+\lvert z_n\rvert.$$ When I first tried proving this result, I tried using induction and the proof was successful. In hindsight though, I'm wondering if there's a simpler way I could have proved it:
The triangle inequality is really stating that the shortest distance between two points is a straight line. Therefore, to prove this we may observe that $$\begin{align}\lvert z_1+z_2+\cdots+z_n\rvert&=\lvert z_1\rvert\cos\theta_1+\lvert z_2\rvert\cos\theta_2+\cdots+\lvert z_n\rvert\cos\theta_n\\ &\leqslant\lvert z_1\rvert+\lvert z_2\rvert+\cdots+\lvert z_n\rvert\end{align}$$ as required, where $\theta_i$ is the difference between the argument of $z_i$ and the argument of $z_1+z_2+\cdots+z_n$.
Is this proof valid?
Thanks for your help.