I want to understand, how i get from the left to the right side in the following inequaltiy:
$$\sup\left\{\lvert f(y_1)-f(y_2)\rvert : \lVert y_i-y\rVert<\nu\right\}\leq \sup\left\{\lvert f(y_1)-f(y_2)\rvert : \lVert y_i-x\rVert<\delta\right\}$$
I can use: $$\nu:=\delta - \lvert x-y\rvert$$ and $$y\in U_{\delta}(x)$$
In particular: How do I get from $\lVert y_i-y\rVert<\nu $ to $\lVert y_i-x\rVert<\delta $ ?
If $\|y_i - y\| < \nu := \delta - \|x-y\|$ then $$\|y_i - x\| \le \|y_i - y\| + \|y - x\| < \nu + \|y-x\| = \delta$$ so the set on the left-hand side is contained in the set on the right-hand side.