I would like to understand a bit better the Arzela-Ascoli theorem
Consider a sequence of real-valued continuous functions $\{f_n\}_{n ∈\mathbb{N}}$ defined on a closed and bounded interval $[a, b]$ of the real line. If this sequence is uniformly bounded and uniformly equicontinuous, then there exists a subsequence $\{f_{n_k}\}_{k ∈\mathbb{N}}$ that converges uniformly.
More specifically I was curious to know if instead of requiring uniform equicontinuity we could just require equicontinuity. It seems to me that using the compactness of $[a,b]$ and the plain equicontinuity we get uniform equicontinuity anyways no?
Yes, when the domain is compact metric (like $[a,b]$) these notions are equivalent; for a proof e.g. see the note here. It's the difference between EQ1 and EQ2 there.