Lemma 3.2.3.label Let $p \in [1, \infty)$, then for each $a, b \ge 0$,
\[(a + b)^{p} \le 2^{p-1}(a^{p} + b^{p})\]
Proof. Since $t \mapsto t^{p}$ is convex,
\begin{align*}\braks{\frac{a + b}{2}}^{p}&\le \frac{a^{p}}{2}+ \frac{b^{p}}{2}\\ \frac{(a+b)^{p}}{2^{p}}&\le \frac{a^{p}}{2}+ \frac{b^{p}}{2}\\ (a + b)^{p}&\le 2^{-1}(a^{p} + b^{p})\end{align*}
$\square$
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