This is an inequality problem which I encountered while solving a metric space problem
let $a,b,c$ be non-negative real numbers satisfying $a\le b+c$.
Then for what positive real values of $n$ is the following also true
$$ a^n\le b^n+c^n$$
I honestly have no idea on how to proceed. I came up with this problem while solving a metric space question. Any hints and suggestions will be highly appreciated.