 Solution

The triangle inequality gives $0<\Vert \mathbf{x}+\mathbf{y}\Vert \le \Vert \mathbf{x}\Vert +\Vert \mathbf{y}\Vert =2$. Taking its negative reciprocal givesIn fact independence excludes the equality case $\mathbf{x}=\mathbf{y}$, so the inequality is strict under the stated independence hypothesis. The weaker requested bound holds for every permitted pair.