Bounding the difference of numbers between 0 and 1 with the same power I would like to prove the following inequality (I guess it holds but I'm not able to formally do it).
Consider two numbers $x,y\in (0,1)$ and the positive real number $\alpha$. Then, can I write 
$$|(1-x)^\alpha-(1-y)^\alpha|\leq c \cdot |x-y| $$
where $c$ is a constant depending on $\alpha$? Thank you! 

Function $f$ defined by $f(x)=(1-x)^\alpha$ is continuously differentiable, therefore for all $x, y$ in $[0, 1]$,
$$|f(x)-f(y)|\leq  |x-y| \cdot \sup_{t\in[0,1]} |f^\prime(t)|$$
Since $f^\prime(t)=-\alpha(1-t)^{1-\alpha}$, the inequality you're seeking is true if $\alpha \geq 1$, with 
$$\boxed{|(1-x)^\alpha-(1-y)^\alpha|\leq \alpha \cdot |x-y|}$$
Note that if $\alpha\in (0,1)$, the inequality is not true. For instance, take $\alpha=\frac 1 2$, and $x=1-h$, and $y=1-2h$:
$$\frac{|\sqrt{1 -x} -\sqrt{1-y}|}{|x-y|}=\frac{1}{\sqrt{1 -x} +\sqrt{1-y}}=\frac{1}{(1+\sqrt 2)\sqrt h}\rightarrow +\infty$$
