# Extension of Arzela-Ascoli Theorem in $\mathbb R\;$ [duplicate]

One version of Arzela-Ascoli theorem can be stated as follows:

Consider a sequence of real-valued continuous functions $\;\{âf_nâ\}_{n \in \mathbb N}\;$ 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 $\; \{â{f_n}_kâ\}_{k \in \mathbb N} \;$ 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?

• It can't be true as stated for $\mathbb{R}$. Take $f$ to be, say, continuous and compactly supported (and not the zero function), and set $f_n(x) = f(x-n)$. Then $f_n$ is uniformly bounded and equicontinuous, but no subsequence converges uniformly. – Nate Eldredge Apr 19 '17 at 17:18
• @NateEldredge What assumptions should 've been added then? I need to derive a result similar to Arzela-Ascoli for $\;\mathbb R\;$ – kaithkolesidou Apr 19 '17 at 17:21