(adsbygoogle = window.adsbygoogle || []).push({}); [SOLVED] Help me prove this inequality

1. The problem statement, all variables and given/known data

The inequality in question is

[tex]|x+y|^p \leq 2^p(|x|^p+|y|^p)[/tex]

for any positive integer p and real numbers x,y.

3. The attempt at a solution

For p=1, it is weaker than the triangle inequality.

Suppose it is true for p, and let's try to show this implies it's true for p+1.

[tex]|x+y|^{p+1}=|x+y||x+y|^p\leq |x+y|2^p(|x|^p+|y|^p)[/tex]

And basically, here I've tried using the triangle inequality on |x+y| but the most "reduced form" I got is I arrived at the conclusion that the inquality was true iff

[tex]|x||y|(|x|^p+|y|^p)\leq |x|^{p+1}+|y|^{p+1}[/tex]

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# Homework Help: Help me prove this inequality

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