Inequality of real numbers with exponent For $a,b>0$ are two real numbers and $p\geq 1$. Is the following inequality true 
$$|a^p-b^p|\leq|a-b|^p\;\;?$$

Let $b\geq a$ then $\>b=a+\delta\>$ and $\>\delta>0$
$$
\\|a^p-b^p|\leq|a-b|^p
\\(a+\delta)^p-a^p\leq\delta^p
\\(a+\delta)^p\leq a^p+\delta^p
$$
but $a>0$ and $\delta\geq0 => (a+\delta)^p=a^p+\delta^p+\delta*x\>(x\geq0)($
Binomial theorem$)=>$
$$
\\a^p+\delta^p\geq(a+\delta)^p=a^p+\delta^p+\delta*x\geq a^p+\delta^p
$$
$=>\delta*x=0=>\delta=0\>$ or $\>x=0=>$
$$
\\|a^p-b^p|\leq|a-b|^p
$$
if and only if
$$
a=b
$$
or 
$$
p=1
$$
