A necessity property of compact subset of Skorokhod space This is the Problem 3.16 of Ethier and Kurtz's Markov Processes Characterization and Convergence

Let $(E,r)$ be complete. Show that if $A$ is compact in $D_E[0,\infty)$ (the skorokhod space with value in $E$), then for each $T>0$ there exists a compact set $\Gamma_T\subset E$ such that $x(t)\in \Gamma_T$ for $0\leq t\leq T$ and all $x\in A$.

As I know given $x\in A$, $\{x(t):t\in [0,T]\}$ is totally bounded, so its closure is compact. Since $A$ is compact in $D_E[0,\infty)$, $A$ is totally bounded, so it has a $\varepsilon$-net for each $\varepsilon>0$. I don't know how to continue. Any help please. Thanks!
 A: I get an answer myself.
Given $T>0$, let $B_T=\{x(t)|x\in A, 0\leq t\leq T\}$. It is sufficient to prove $B_T$ is relative compact. Then $\Gamma_T=clo(B_T)$.
Let $\{\alpha_n\}\subset B_T$ be a sequence in $B_T$, then for each $\alpha_n$, there exist $x_n\in A$ and $t_n\in [0,T]$, such that $\alpha_n=x_n(t_n)$. Since $A$ is compact, by take subsequence we can assume $x_n\to x$ in $D_E[0, \infty]$. Since $\{t_n\}$ has convergence subsequence, by take subsequence we can assume $t_n\to s\in [0, T]$. Given $\delta>0$, by the Proposition 3.5.3.(c) of the same book, there exists $\{\lambda_n\}\subset\Lambda'$ such that
$$\lim_{n\to\infty}\sup_{0\leq t\leq T+\delta}\mid\lambda_n(t)-t\mid=0$$
$$\lim_{n\to\infty}\sup_{0\leq t\leq T+\delta}r(x_n(\lambda_n(t)),x(t))=0$$
where $\Lambda'$ is the collection of strictly increasing functions $\lambda$ mapping $[0,\infty)$ onto $[0,\infty)$. By take subsequence we can assume $\lambda_n(t_n)\to s+$ or $\lambda_n(t_n)\to s-$.
For the first case, 
$$r(x_n(t_n),x(s))\leq r(x_n(t_n),x(\lambda(t_n))+r((x(\lambda(t_n)),x(s))$$
For the second case,
$$r(x_n(t_n),x(s-))\leq r(x_n(t_n),x(\lambda(t_n))+r((x(\lambda(t_n)),x(s-))$$
They both converge to $0$ by the cadlag property of $x$ and the limits we get above. So the problem has been proved.
