I want to prove, that function is Lipschitz continuous. I'm stuck on one step. I have the solution, but I want to solve it $\textbf{without}$ derivative, so just using definiton. So let
$f:[0,1]\to \mathbb R,\ f(x):=\sqrt{x-\sin(x)}$.
$\textbf{Proof:}$ So let $x,y \in [0,1].$ Let $\epsilon>0$. Choose $\delta=...$.
Then $\forall \ x,y \in [0,1]$ with $\lvert x-y \rvert<\delta$,
we have
$\lvert f(x)-f(y) \rvert=\lvert \sqrt{x-\sin(x)}-\sqrt{y-\sin(y)}\rvert=\Bigg \lvert \frac{x-\sin(x)-y+\sin(y)}{\sqrt{x-\sin(x)}+\sqrt{y-\sin(y)}} \Bigg \rvert=\Bigg \lvert \frac{x-y+\sin(y)-\sin(x)}{\sqrt{x-\sin(x)}+\sqrt{y-\sin(y)}} \Bigg \rvert=$
and now I'm stuck. What have I to do? How can I evaluate this in order to find a Lipschitz constant K?