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.