proof of $\lim_{k\to\infty} x_{n_k} = \lim_{n\to\infty} \sup x_n $
$a_{n_{k-1}+1} = \sup\{a_{n_{k-1}+1},a_{n_{k-1}+2},a_{n_{k-1}+3},a_{n_{k-1}+4},....\}$, or $a_{n_{k-1}+1} = \sup\{a_{n_{k-1}+1},a_{n_{k}+1},a_{n_{k+1}+1},a_{n_{k+2}+1},....\}$ Could you tell me which one is right?, and I also don't understand why $a_{n_{k-1}+1}\ge a_{n_k}$. Thank you in advance.
The former equlaity is right. You just have to put $n_{k-1}+1$ in place of $n$ in the definition of $a_n$. Also, since $n_{k-1}+1 \leq n_k$, the set where the supremum is taken is larger for $n_{k-1}+1$. That is,$$\{l\geq n_k\} \subset \{l\geq n_{k-1}+1\}$$