I am trying to understand a proof. The statement is
The proof begins with:
I have some questions about it:
- ANSWERED Since I am Italian, when it is said "the supreme is ATTAINED" what does it mean? Because $\xi\in\mathbb{R}$, so how couldn't be possible it?
- $|\xi_\nu|\rightarrow \infty$ means I am considering an increasing sequence?
- What is the aim of the Step 1, that is why must I proof that there is an $\xi$ for $H$ and that exists $R$ so that $|\xi|<R$?
- The fact that $\frac{w|\xi|}{|\xi|}>R+1$ derives from the hypothesis of the limit, but where does the following inequality comes from?
- ANSWERED I don't understand the Step 2, in fact I tried to find the other inequality but I only find out that
$(v-v')\xi'+f(x',u',\xi')-f(x,u,\xi') \leq H(x,u,v)-H(x',u',v') \leq (v-v')\xi+f(x',u',\xi)-f(x,u,\xi)$
Hope there will be someone who can help me, I have to write something on it and I would like only to understand it well. Thank you :)