Extension of the Triangle Inequality

[Caspian]

Pretty much knows the triangle inequality.
\left| a + b \right| \le \left| a \right| + \left| b \right|

I was reading a source which asserted the following extension of the triangle inequality:
\left| a + b \right|^p \le 2^p \left(\left| a \right|^p + \left| b \right|^p\right)

This is bugging me because I can't figure out how to prove or disprove it. It's sensible enough that it might actually be true... but want to know for sure whether it's true.

I tried proving it by induction, but it got really messy...

Is this inequality true? Or, is it wrong? Thanks!

[joshyxc1979]

when you tried induction, did you try the binomial theorem?

[losiu99]

It follows from convexity of xp for p > 1.

(\frac{1}{2}|a|+\frac{1}{2}|b|)^p\leq \frac{1}{2}|a|^p+\frac{1}{2}|b|^p


(|a|+|b|)^p\leq 2^{p-1}(|a|^p+|b|^p)