 Reverse triangle inequality for future timelike vectors

Use signature $(+---)$ and future-pointing timelike vectors $U,V$. Their time components exceed their spatial magnitudes, so the same holds for their sum by the Euclidean triangle inequality. In the rest frame of $U$, write $U=(a,0)$, $a=\sqrt{g(U,U)}>0$, and let $b=\sqrt{g(V,V)}>0$. Then $g(U,V)=a{V}^0\ge ab$. Expanding the squared norm of $U+V$ proves the displayed inequality. Equality requires $V$ to have zero spatial component in this frame, equivalently $V$ is a positive scalar multiple of $U$. The shared future time orientation is essential.