Let $(X,d)$ a metric space and $f:X→\mathbb{R}$ Lipschitz function. Define for each $n\in\mathbb{N}$ a function $ f_{n}(x):=\inf_{z\in X} \{f(z)+n\cdot d(x,z)\}$. It is clear to me that for all $x\in X$, we have $$ \inf_{z\in X}\{f(z)\}\leq f_{n}(x)\leq f(x) $$ and then $f_{n}$ is bounded. But have a question:
Why is $f_{n}$ n-Lipschitz? confess that I am having difficulties with the accounts. Let $x,y\in X$: $$|f_{n}(x)-f_{n}(y)|=| \inf_{z\in X} \{f(z)+n\cdot d(x,z)\}-\inf_{z\in X} \{f(z)+n\cdot d(y,z)\}| $$ and I confess that I'm having trouble reasoning this out.
