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Show that for each $r> 0$ $$\mathbb{E} |X+Y|^r \leq c_r (\mathbb{E} |X|^r + \mathbb{E} |Y|^r),$$

where $c_r$ is a constant given by

$\begin{equation} c_r = \left\{ \begin{array}{ll} 1 & \mathrm{if\ } 0 < r \le 1 \\ 2^{r-1} & \mathrm{if\ } 1 < r \end{array} \right. \end{equation}$

I've tried to use other inequalities for the proof of this one but I still get stuck for the case of $2^{r-1}$.