I have a question on Ascoli Arzela theorem.
In Royden's Book Theorem is given as follows.
Let $\digamma$ be a equicontious family of functions from a separable space $X$ to a metric space $Y$. Let $<f_n>$ be a sequence on $\digamma$ such that for each $x$ in $X$ the closure of the set $\{f_n(x): 0\leq n< \infty \}$ is compact. Then there is a subsequence $<f_{n_k}>$ that converges point wise to a continuous function $f$, and the convergence is uniform on each compact subset of $X$.
But when I'm solving problems using Ascoli- Arzela theorem,To show functional space is compact, what I need to show is sequence of functions on compact interval is uniformly bounded on a compact interval and sequence of functions are equicontinuos.
I'm confused with the Theorem in Royden's Book. Is it similar to what I have been used in problem solving.