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[SOLVED] Help me prove this inequality
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.
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]
Homework Statement
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.
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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