Triangle inequality raised to fractional powers Does the inequality $|x + y|^\alpha \leq |x|^\alpha + |y|^\alpha$ where $\alpha \in [0, 1)$, hold for all $x,y$ ??

For the $x\ge 0,y\ge 0$. $f : t \mapsto (t + x)^{\alpha} - t^{\alpha}$ is decreasing function on $[0,+\infty)$ (take the derivative $f'(t) = \alpha ((t+x)^{\alpha -1} - t^{\alpha - 1}) < 0$ since $\alpha < 1$) so $$|x|^{\alpha} = f(0) \ge f(y) = |x+y|^\alpha - y^\alpha$$
