For any norm, $\|x+y\|\le\|x\|+\|y\|$. For an inner-product norm, expand $\|x+y\|^2$ and bound the cross term using Cauchy–Schwarz: $\operatorname{Re}\langle x,y\rangle\le\|x\|\|y\|$. Taking square roots gives the result. Geometrically a direct path is no longer than a two-segment path.
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