One version of Arzela-Ascoli theorem can be stated as follows:
Consider a sequence of real-valued continuous functions $\;\{ f_n \}_{n \in \mathbb N}\;$ defined on a closed and bounded interval $\;[a, b]\;$ of the real line. If this sequence is uniformly bounded and equicontinuous, then there exists a subsequence $\; \{ {f_n}_k \}_{k \in \mathbb N} \;$ that converges uniformly.
I am trying to extend the theorem for sequences of real-valued functions defined on $\; \mathbb R\;$. One hint I've got , is to use a diagonal argument...but I'm not very familiar with this.
EDIT: I need to derive that a sequence of real-valued functions defined on $\; \mathbb R\;$ which satisfy the conditions of boundedness and equicontinuity on a closed interval of $\; \mathbb R\;$, has a uniformly convergent subsequence on compact intervals.
How do I proceed?
I would appreciate if somebody could enlighten me about this. Any help would be valuable!
Thanks in advance!!!