Yet another inequality: $|a+b|^p<2^p(|a|^p+|b|^p)$ Let $a$ and $b$ be real numbers and $p>0$. What is the best way to prove that $|a+b|^p<2^p(|a|^p+|b|^p)$? 

If you divide $2^p$ on both sides, then you get $(|a+b|/2)^p < |a|^p + |b|^p$. Can you see why that would be true? 
