I need to prove that $|x^{1/n}-y^{1/n}|\leq c|x-y|^{1/n}$, where $x,y\in [0,+\infty)$, $n\in\mathbb{N}-\{0,1\}$, and $c=2^{(n-1)/n}$.
I tried a lot ways for proving cases $n=2$ and $n=3$, but are quite different, I don't know how to prove the general case.
I've done case $n=2$ in the following way. I was prove that $|\sqrt{x}-\sqrt{y}|\leq \sqrt{|x-y|}$ squaring both sides when $x\geq y$.
The case $n=3$ I've done in a similar way. I was prove that $|\sqrt[3]{x}-\sqrt[3]{y}|\leq \sqrt[3]{|x-y|}$ cubing both sides, but is quite different for the case $n=2$. I don't know if it is possible prove using induction with a binomial expansion. I only prove both cases $n=1$, $n=2$ using $c=1\leq 2^{(n-1)/n}$, therefore I proved a better inequality for the first two cases.