proving that $f(x) = x^s$ is holder continuous with holder exponent s
Assume $x>y$.
$$x^s-y^s \leq C(x-y)^s$$
$$\Leftrightarrow x^s \leq C(x-y)^s+y^s$$
Claim: this holds for all $(x,y)$ when $C=1$. Proof: Because $0<s\leq 1$, for all $a,b \geq 0$
$$(\frac{a}{a+b})^s+(\frac{b}{a+b})^s \geq 1$$
$$\Leftrightarrow a^s+b^s \geq (a+b)^s$$
Now set $a=x-y$ and $b=y$.