Requirement of closed and bounded set $[a,b]$ in the Ascoli theorem In Wikipedia, the Ascoli theorem requires the functions to be continuous on the
closed and bounded interval $[a,b]$. However, in the proof given in the book 
"Theory of Ordinary Differential Equations" by Coddington and Levinson
(Tata MaGraw-Hill Edition 1972) (given at page 5, Ascoli Lemma) the authors only require that the interval be bounded. The lemma is:-
On a bounded interval $I$, let $F=\{f\}$ be an infinite, uniformly bounded, 
equicontinuous set of functions. Then $F$ contains a sequence $\{f_n\}$, 
$n = 1,2,\cdots,$ which is uniformly convergent on $I$.
(The authors mention in the text before the lemma that $I$ denotes an open 
interval.)
Reading through the article on Compact Space on wikipedia, I get a feeling 
that the interval should be compact. However, the proof in the above book also
seems to be correct.
If someone has access to this book (as it is not available on google books), can
you please clarify.
Thanks in advance. 
 A: The authors define "equicontinuous on $I$" to mean "uniformly equicontinuous on $I$".  That is, for any $\epsilon>0$ there is a $\delta>0$ such that for any $f\in \cal F$ and for any
$x$, $y$ in $I$,
$$|f(x)-f(y)|<\epsilon\quad\text{ whenever }\quad|x-y|<\delta.$$
The theorem you are referring to does give uniform convergence (you omitted this in your post). And, indeed, with the assumption of uniform equicontinuty   and uniform boundedness, the assumption that $I$ be closed is not needed.
What is needed is that the interval be totally bounded. If you examine the proof in Coddington and Levinson , you'll see that's what they are using.
See here for a different proof.
 Incidentally, some authors say $\cal F$ is  equicontinuous over $I$ to mean that given  $x\in I$ and   $\epsilon>0$, there is a $\delta_{x,\epsilon}$ such that $|f(x)-f(y)|<\epsilon$ whenever  $|x-y|<\delta_{x,\epsilon}$ and $f\in\cal F$ . One can show that if $I$ is compact, then this notion of equicontinuity implies the notion of uniform equicontinuity above.
