In the proof of a compactness theorem (Rellich-Kondrachov) in Evans's Partial Differential Equation, the following argument is made:
I'm not able to fully understand step 7.
Here are my questions:
- Would anyone elaborate how the diagonal argument is used to give the desired subsequence? (To be precise, assuming Step 6 is done, how to get Step 7?)
- How should one understand the notation $\limsup_{j,k\to\infty}$ appropriately?
Let $F_1=(f_1(n))_n$ be a sub-sequence of $\mathbb N$ with $\lim_{n\to \infty} \sup_{n\leq j<k} \|u_{f_1(j)}-u_{f_1(k)}\|<1.$
Take $G(1)=f_1(n_1)$ such that $\forall j>n_1\;(\|u_{f_1(n_1)}-u_{f_1(j)}\|<1).$
Let $F_2=(f_2(n))_n$ be a sub-sequence of $F_1$ with $f_2(1)=f_1(n_1),$ such that $\lim_{n\to \infty}\sup_{n\leq j<k}\|u_{f_2(j)}-u_{f_2(k)}\|<1/2.$
Take $G(2)=f_2(n_2)$ with $n_2>1$ ( i.e. $G(2)>G(1)\;$) such that $\forall j>n_2\;(\|u_{f_2(n_2)}-u_{f_2(j)}\|<1/2).$
Let $F_3=(f_3(n))_n$ be a sub-sequence of $F_2$ with $f_3(1)=f_2(n_2),$ such that $\lim_{n\to \infty}\sup_{n\leq j<k}\|u_{f_3(j)}-u_{f_3(k)}\|<1/3.$ ET CETERA.
Consider the sequence $(u_{G(n)})_n.$
This is with the assumption that for any $\delta>0$ and any sub-sequence $F$ of $\mathbb N$ there is a sub-sequence $F'=(f'(n))_n$ of the sequence $F$ such that $\lim_{n\to \infty}\sup_{n\leq j<k}\|u_{f'(j)}-u_{f'(k)}\|<\delta.$